If I had to guess I would say it is either dairy related or laundry related.
My son also had this and it covered his entire body plus his little head. It was awful! It was also related to dairy. It took me about two months to figure it out and I switched him to soy formula (I was previously pumping and supplementing with regular formula) His problems immediately started clearing up (although with dairy intolerances it could take up to a month or so to completely be free of symptoms) and it made all the difference. He now is able to drink and eat dairy products but when his eczema is really bad I try and cut it down a bit.
Get a Rx steroid cream to use a couple times a day. Other than that I really like Aquaphor but it's pretty greasy so I usually only put it on at night. I use all free and clear type soaps and shampoos, for the most part and even at 3 I still don't use any sort of fragrance lotion on him.
My daughter on the other hand, had similar symptoms and it was directly related to fabric softener. I never put it in her laundry but I did put it in ours and even when we would just hold her or she'd lay on a blanket or something she would break out. I stopped putting it in anyone's laundry and her skin cleared right up. I still don't use it any of our laundry and you know what, I don't notice a huge difference. I will say that I do miss that "fresh" smell that it gives but I don't feel it did much for any other thing. I use vinegar in all my wash loads and that helps.