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No... the door in the lich's study is opened with the key you can get from Adrianna. That key you do not need since the lock can be picked... although it takes a skilled character.
It has been a while, so I am not sure about this, but I think the key for the vault is called the vault key. Where have you found the skull key in the first place and have you already used it somewhere else? Skull Castle contains three or four keys, and so far as I recall they only fit one place each.
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