The Skull Key is used to unlock the Skull Cavern. The Skull Cavern resides in a cave in the top-left hand corner of the Calico Desert, accessed by the bus. The Skull key is found if you reach the 120th level of the mine which is also the bottom of the mine.
When you get the Skull Key, you can play the Junimo Kart arcade machine in The Saloon. It is located in the far right room next to the Journey of the Prairie King arcade machine. The reward for beating the Progress Mode of the game is a copy of the game in a console version. This copy can be placed at the farm's house.
You can find the Skull Key in your 'Wallet' in the The Player's Menu within the Skills tab.