Skipping the Skull Hammer requires being able to skip all instances of the hammer being supposedly required. This includes skipping the Phantom Ganon fight in Forsaken Fortress 2, the hamer peg at the top of FF2, the Helmaroc fight, a few rooms in the Earth Temple, the Private Oasis sewers and entering Gale Isle.
Skipping using the Skull Hammer in FF2 is precisely what is called the Early Master Sword sequence break, click here for more information and the many ways to do it.
Helmaroc can only be defeated using the Skull Hammer. As such, you have to skip Helmaroc if you wish to skip the hammer. See the Helmaroc King page for all the ways to skip him.
The first required use of the Skull Hammer in the Earth Temple is right in the main room with the big statue blocking the access to the basement. In this room, you are supposed to use the hammer on a peg in order to unlock the door on the right wing, which allows you to get the small key you are supposed to use on the left wing to progress towards the mirror shield. There are two ways to skip this hammer peg: one is to obtain the Mirror Shield early using the Tingle Tuner, the other is to get a small key in the basement and use it on the left wing door instead of the intended one.
Once you have defeated the chus in the previous room, use Medli to reveal the chest. Now get dry storage off of one of the pillars in the room and get chest storage. After opening the chest, your screen will turn black, except for a few key features such as Link and Medli's eyes and the ring of light. Using these features, along with your mini-map, pick-up Medli and navigate your way to the Statue room. Note: It is very important that you reveal the chest before you get dry storage, otherwise you will store the cutscene that reveals the chest, and will be unable to open it until you in some way reset the dungeon (such as dying or saving and quitting).
Once you're in the statue room, use Medli to glide over to the statue's left (your right) arm. After you have landed on it, you can drop Medli and maneuver yourself up to the statue's elbow. Wait for the effects of the purple mist to wear-off (if they have not done so already), and use a jumpslash to get on the statue's earring and then another to get onto its head. Sidehop onto the crest on top of the head and then glide over to the ledge with the door. Getting on the crest is not necessary to reach the upper ledge, however it does eliminate the need for leaf pumping--saving some magic.
If you have not done so already, call Tingle and reduce your health to 1/4th heart before entering the next room. It's crucial that you lower your health before entering the next room as it will interfere with the following set-up if you do not. Enter the next room, C-Up and align the right side of the B-Button with the black corner line on the wall. Turn around, sidehop left, jumpslash, backlfip, lay down a Bomb, slash it without L-Target and start the zombie hover.
If your followed the above set-up correctly, the edge of the door frame will help to clip you out-of-bounds. As you are hovering, put a blue ting on B on the Tingle Tuner. Once you have gained significant height, cue the ting and mash the deku leaf right before you get healed. Note: There is a good degree of timing as to when you start mashing the deku leaf. If you begin mashing too early you will exhaust you magic meter and may lose too much height before the ting heals you. If you start mashing too late the ting will heal you before you can get the leaf pulled out and you will jumpslash into the void. This is partially why you are selecting a blue ting--so as to refill both you health and your magic.
After you've pulled out the deku leaf, glide over to the loading zone that leads to the mirror shield room (you may have to use a few leaf pumps to reach it). Note: Once you have obtained the mirror shield you must save-warp to return to the beginning of the dungeon, otherwise you are trapped as you never defeated the enemies to lower the wall in the chronologically preceding room.
This following video shows the Setup that is used for the zombie hover:
Once you have the Mirror Shield, you can progress through the 2nd part of the dungeon.
For this method, you will need to skip past the Statue and access the basement without the Mirror Shield but with Medli. This can be done several ways (see here).
For exemple, if you have the Tingle Tuner, you can zombie hover to the upper ledge and use a red ting to heal once you're up there, and then ledge clip with Medli:
Or you can roll clip OoB, get behind one of the door frames on the side of the room, pick up Medli and put her down OoB. Then use the Command Melody to make her fly behind the statue, and leaf your way to her once that is done:
Or you can clip with Chest Storage from the back of the Statue.
Once in the basement with Medli, you can use the Earth God's Lyric and the Command Melody to progress normally through the 2nd part of the dungeon. When you obtain the small key in the basement, savewarp to get put back at the begining of the dungeon. Use the small key you got in the basement to unlock the door on the left wing. This skips the first hammer peg. You will just need to perform any method of the B1 Small Key Skip later to avoid the door you were intended to use that small key at.
If you haven't used every key in Dragon Roost Cavern, the Forbidden Woods or Tower of the Gods, you can use one of these keys in the Earth Temple by transferring it. To transfer a small key to another dungeon, store the chest containing the small key, and then leave the dungeon without reloading the area. The small key will be registered as being in the next dungeon you enter. Doing this to get an extra key in the Earth Temple skips the first hammer peg.
The Second Hammer Peg is located in the room with the grate and the elephant statue in the basement. This hammer peg is used to bring light in the room, light that is required in order to destroy the elephant statue. However, it is possible to skip doing this thanks to chest storage.
After getting the Mirror Shield, go to the third room in the dungeon, and reveal the treasure chest. Preform storage and open the chest. Then, navigate to where the peg is and perform the B1 small key skip with Chest Storage. This also allows you to bypass the locked door whose key you used to get to the Mirror Shield if you did method 2. Then, jump and glide onto the pillar that is to the left of the statue. Backwalk into the wall and grab the ledge to ledge clip OoB. Glide over into the doorway. From there you can open the door and progress past the statue without breaking it.
Note that you can place Medli inside the statue with chest storage if you need her past this point in the dungeon. Once behind the statue you can call Medli with the command melody. Pick her up and open the door to proceed.
With Chest Storage, from the boat, you can leaf and land on one of the pillars on the eastern side of Gale Island. You can then climb up it and leaf to a higher ledge on the island. With Chest Storage, you can simply clip through the island somewhere on the back and enter the loading zone without the hammer. You can then proceed by learning the Wind God's Aria.
Make sure the wind is blowing north, call Tingle and zombie hover above Gale Island. Mid-hover, equip a Red Ting to B on the Tingle Tuner. Alternatively, you can hover to the hearts on the island to the side of Gale Island. Once you are above where the wind is blowing from, use the ting to heal yourself. Walk over to the ledge on the right and backwalk until Link grabs the ledge and clips into the rock face. From here drop down and immediately use the Deku Leaf to glide over the loading zone.
Note: There is also a peg that you would normally need the skull hammer to activate in order to unbar the door in the Mini Boss room so that you can exit. However this requirement is easily overcome by save warping back to the first room of the dungeon, or by escaping the room with chest storage.
This is one of the simpler times in skipping the Skull Hammer requirements. Sidle part-way into the gates and roll the rest of the way through. This is the only place in the sewers where the skull hammer is normally a requirement, as well as the only time when skull hammer requirements are impeding skipping it in the Triforce-collecting sequence.
There is a faster way to skip the hammer in the Cabana sewers, involving the hookshot and a L-slide clip.