Is it possible to change your single-player skin?

When you change your skin, it affects both single player and multi-player.

If you don't see your skin in single player, make sure you're online so it can download the skin.

Yes it is possible. Multi-player and single-player modes use the same skin. Create or download your skin, then upload it by logging into minecraft.net, and selecting the Profile link.

Since there's no reflective surfaces in Minecraft, the only way to see your skin in single player is to go to third person view by pressing F5 (and again to switch back to first person view) or simply opening the inventory.