Is there any meditation technique to be aware of what you hear?
Yeah, you could probably make that happen.  What you’re looking for is a sense sensation that has some stability.  Visually, you would use something like a kasina, but I see no reason why a sound kasina couldn’t also work provided that you’re diligent about it.  As to what this sound kasina would be made up of, you’d want something that was consistently present.  A dripping faucet would probably work, but I’d be more inclined to use something like that background hum you were talking about.  You could also plug your ears and listen to the rushing sound of your blood in your ears.

Remember though, the whole point of a kasina is to generate a counterpoint sign.  When that arises, you would abandon whatever sound you were working with.