If you want to keep it looking good don't use an abrasive brush or steel wool. Just use water and dish detergent for regular cleaning and towel dry. If it is getting dark spots use ammonia. Make sure you rinse well afterwards and dry it completely.
If that still doesn't do the trick, there are specialized stainless cleaners. If you use an abrasive brush it will leave nasty scratches all over it.
OOOO (four ought) steel wool will not scratch the steel. The rust you are removing is more abrasive than the steel wool. That is why you use it wet with solvent or light oil to flush away the rust you are removing.
This method works well with blued steel too.
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"Robo is right" Fuzzball