Is it possible to skip past night in Minecraft single player?

Yes, use a bed. Build it like this (spoiler, hover above the grey box):

Three times wool above three times wooden planks

Also, you can use a Furnace as temporary light source and as by-product obtain charcoal:

Get some wood, build a workbench and there a wooden pickaxe. Dig down until you reach stone, get 8, build a Furnace, use unprocessed wood as input and wooden planks as fuel. You obtain light plus charcoal for torches.

This is how you make a bed:

And this is how you make charcoal:

Well there are four most popular ways.

1. Make a bed... As Tobias suggested
2. Wait it out with the window open
3. Open your inventory ('e' or 'i'), then switch window. This stops the game from pausing.
4. Hacks :P

If you have a bed already you can sleep during the night with following construct.

      __
     xxx
     x
     x
gggggggggggg

Where g stands for ground, x is any material that can float (so no sand or gravel) and __ is your bed. When you build it like this you are save for to sleep in that bed (granted there isnt a ridge nearby where a skeleton could start shooting you.