Fish Oil supplements thrown in with food can work wonders in helping skin
and coat- I give my dogs
3-5 capsules a day, (get the cheapest kind at Costco - just as good/better as high dollar "pet" brands).
My vet told me 90% (or more) of allergic-type symptoms come from diet. Read the labels on your dog food- corn, wheat, rice, (grains) can cause problems.
A "limited ingredient" dog food, especially one that is grain-free would probably make a big difference.