A bucket of soapy water and a scrub brush is an excellent way to mechanically clean a crate. Adding some
laundry borax to the bucket of hot soapy water is even better. This won't disinfect, but it will
clean -- and that's all that may be needed for a crate used by the same healthy dog. The borax also deodorizes (removing doggie smells).
In crates used by foster dogs, I do a scrubbing, but after that we also spray a 10% bleach solution and let it sit on the crate surfaces. This is for decontamination purposes, since they tend to come out of shelters with all sorts of viral and bacterial crud (kennel cough, etc.). After the bleach has had time to work, we rinse and air dry in the sunshine.
ETA: you mentioned "dog crate mat," OP -- I think a lot of us are assuming that you mean the black plastic crate pan in the bottom? If you are talking about a soft cloth mat, I'd follow label directions regarding laundering. If the cover (or whole thing) is washable, I wash all my dog bedding in hot water with laundry detergent and a scoop of Oxyclean, and I dry on "high heat" to make sure any invisible critters (dust mites, etc.) get killed. If it has a rank doggie odor, laundry borax is your friend -- follow box directions to add to the load or do a pre-soak. I'm about to do that with a comforter that a shelter dog with a bad infection was wrapped in when I pulled him--the comforter still stinks of infection after being washed, so it's going in a laundry sink of borax solution to get the smell out.