If you're going to a good butcher, yes, the outside of the beef does rot. After which the butcher cuts off the rotted bits, leaving behind some fantastic tasting steak.

However, if you buy an aged steak from a supermarket, they'll have done what's known as "wet ageing". Basically they vacuum pack it, and then leave it to sit for 30, 60, 90, or whatever days. During which nothing much happens because oxygen is needed to make the ageing process happen, but it does mean they can sell you a "30 day aged" steak without having to throw away any meat.