You don't skin a cat, you swing it. And it is not a feline animal. It is a cat o'nine tails, a horrible nine-tailed whip used to punish sailors and leave their backs bleeding. This was done in the open air, for below deck there would not have been room to swing the cat, and in any case the rest of the crew were required to witness the punishment.
This origin of the phrase has been questioned, but it is the only explanation that convinces.