Best way to clean stainless pans

So I have just bought a new stainless pan to give it a go and while I know how to get it to the correct temp to make it non stick what do you all do about all the leftover oil and butter in the pan (outside the cooking area of steak or eggs) that seems to just cook into a brown sticky paste on the sides. Is the only way to get this off once finished to just scrub the crap out of it with a scourer? I can get it almost like brand new clean this way but it's also starting to scratch the surface.

I'm ok with it if this is the case but just curious as to what everyone else is doing to clean the brown sticky mess? I am looking to redo my kitchen and would like to go the professional style with pans hanging on the wall but everything needs to be spotless if this is the case.