What makes oil stick to the pan so bad that it is so difficult to wash out?
As I understand it, when you combine high heat and vegetable oil you get a fairly stable polymer (much like a plastic or resin).  The polymer bonds with the surface (which is porous), and results in robust surface. This is what we use to season cast iron cookware, but it's less desirable on stainless and other lighter coloured hardware.

You can prevent the coating from forming by never allowing he oil to maintain a high temperature for long periods by itself.  Keeping food in contact with the pan will both help keep the temperature down, and reduce the chance of the oil bonding with the steel.

Cleaning the oil off once it's formed is usually pretty simple: add water (or soda water) and heat for 10-15 minutes. Often the oil will wipe out (or need mild scrubbing with an appropriate scrubbing pad).  You should not have to use steel wool (or a caustic cleaner), both of which can damage the smooth surface of many pans.