Fish oil is very important. It can be a human-grade capsule (labeled with the "USP" means a third party checked for purity) -- for example, Costco capsules work for lots and lots of dogs, but it takes a good month to see a benefit. You can buy much more expensive forms with higher levels of EFAs, if your budget allows. (e.g., Nordic Naturals makes a pet supplement in a blue glass bottle, which gets stored in the fridge).
The Vitamin E is worthless if it's synthetic E. It needs to be a natural form the body recognizes, preferably with the full spectrum of mixed tocopherols. Any product labeled "mixed tocopherol" should be fine, but you won't find them at big box stores or drug stores in the U.S. -- they're available from natural supplement stores, Whole Foods, health stores, or online.
A dry itchy patch could be lots of different things though -- from ringworm (fungus), to bacteria, to parasites (like mites). I agree that I'd start with a skin scrape. That's a procedure that normally costs under $25, plus the exam fee. They scrape off a layer of skin with a sterile razor blade, mount the scraping on a slide, then look under the microscope to see what's there.
As for grain free, IMHO that fad is a kibble marketing gimmick. If you're feeding kibble, you're feeding starch because the extrusion process won't work without it. It needs the starch as a binder. The grain-free types just substitute potatoes (white or sweet), tapioca, or some other starch in place of a grain--and the substitutes might not be digested as easily! Rice and oats are both digested pretty well by most dogs. Unless there's a food allergy, I don't think you're gaining a lot by paying more for a grain-free kibble. Now, if you're paying more to get the dog off of kibble and onto an alternative diet, then you're potentially stepping up to something much better.
That said, as kibble goes, I do like foods like Fromm 4-Star that have both grain and grain-free options and are designed to be fed on rotation, with no transition. You get a broader nutrient profile by rotating through the entire line. My Fromm-fed dogs have thick, lush coats with a fish oil capsule a day. These are dogs that have no allergies, so it works for them.
If you think you have a dog food allergies, then you would be looking at an elimination diet (lots of threads about it), and a limited ingredient-labeled kibble (Wellness Simple, Natural Balance LID, and on an on with competitors now). I wouldn't go that route yet -- what you're describing doesn't sound like an allergy (and statistically most skin issues aren't likely to be allergic responses in dogs).