There's more than one way to skin a cat.

I've cut my finger at Burning Man while taking down our shade structure - 8 days on the playa without a bath. My skin was coated in playa powder, sun block, and skin moisturizer. The key was to use a band-aid big enough to wrap around my finger and stick to itself. shrug - cleaning the area around the skin would have been preferable, but I wouldn't use alcohol for that. Soap and water works fine, if you've got it.

If you have a wound on an area the band aid won't reach around, and you're out of ways to clean the skin, either run a piece of tape around the limb (I've got yards of surgical tape in my kit), or put a gauze pad on the wound and tie it with fabric.

Oh, and crazy glue works, too. Just glue the cut shut.