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This sounds like something for a dr to look at, but I just wanted to answer your question that no, it does not sound like eczema. Eczema is red, VERY itchy, patches, not bumps, and after scratching can become infected and be oozy, not dry. At least, that's my experience with it.
Could it be this? http://www.nlm.nih.gov/medlineplus/ency/article/001462.htm
My daughter has this on the back of her upper arms.