This is a common piece of trivia, and the answer is... (drum roll, etc.):
The skin isn't just a covering for the body (although that function is absolutely vital) - it's a fully functional organ that also regulates body temperature, prevents evaporation of our bodily fluids, protects the us from infections, and acts as a waterproof shield (so that our cells don't burst in the rain or the bath, for instance). And we're only just getting started.
But how big is the organ actually? That obviously depends on you. If you really want to know, you could take a billion measurements, apply some hefty calculus and come up with an answer.
Or you could just go to this page and punch in some numbers. (I prefer the second approach.) The page allows you to use several of the commonly used formulas to calculate your body surface area. These formulas are obviously only guides - they make certain assumptions and extrapolate from limited data. But given the alternative, they're most welcome!