I, too, recommend home-cooked for an elimination diet trial. I have lots of experience with food allergies, seeing as how both my pets have it and I deal with it often. It's rather frustrating, but can definitely be managed.
My dog started off being allergic to processed chicken (commercial food), but was fine with cooked and raw chicken. Over time, she became sensitized to fresh chicken as well, but it goes to show that how the meat is cooked can definitely change things.
My cat is the same way. He cannot eat chicken in kibble/canned, but can eat it in cooked and raw form. Moreover, there is something in commercial food (most likely additives) that makes him itch that is not evident in his home-cooked diet. He can eat commercial food that contains the EXACT same ingredients as a home-cooked batch and itch on it while no issues with the home-cooked stuff.
BTW, both of them are on homemade diets now.
An elimination trial is most effective when using a novel protein, so I would actually avoid chicken and choose a protein that your dog has NEVER had before, e.g. rabbit, venison, bison, etc. Keep at the diet (with no treats or supplements containing any other protein aside from the chosen one) for 8 weeks and if the skin condition improves, then it most likely is a food allergy. If no improvements, then you might want to start looking at other possible issues, but of course, I would first have a vet evaluate and examine the skin and fur to rule out the easier-to-deal-with stuff before starting an elimination diet. The trial is a pain and you must remember not to give any treats or food that is different from the protein being used in the trial or all your efforts will go to waste and you will then need to start all over...but this is the most accurate way to determine food allergies. Good luck!
P.S. If home-cooked is totally out of the question, I would probably do the elimination diet with
Natural Balance. They sell dog food with very short ingredient lists (one protein + one carb) and each flavor/protein comes in both kibble and canned.