An easy solution to your emergency bandaging needs, without the pain of sticky tape in your arm hair.
Step 1: Scrape up your arm
This is probably harder than it sounds, because you need to remember to get your hands/wrists out of the way when you fall. Falling while holding on to something (dinner, your child, bike handlebars, priceless art) can allow you to get a proper scrape on your forearm, elbow, or upper arm.
This instructable utilizes a fine example of road rash on the ulnar side of the left forearm just below the elbow.
Step 2: Apply first-aid
Clean your scrape: remove foreign objects, staunch blood flow, and apply the best disinfectant you have at hand. Once this primary care is done you're ready for a bandage.
If your injury is really manky and you have access to proper medical supplies, you may choose to use a more traditional bandage for the first day or two before moving on to the sock technique.
This injury was dressed with a proper first-aid kit at the site of the incident, thus the nasty tape-marks on the inside of the arm. After painful tape removal in the shower the next morning, we decided to bring out the sock.
Step 3: Acquire a clean sock
Steal a clean sock from your significant other. Make sure it's at least plausibly near the end of its life; wear near the toe is best, as you're going to cut it off anyway. It should be large enough to fit around the relevant parts of your arm, but tight enough to stay put. Try it out on your un-injured arm if necessary.
If you're a paranoid freak you could certainly boil, bleach, or otherwise sanitize your sock; of course, you're probalby not the sort of person to be making a sock bandage in the first place.