It sounds like eczema. I would take her to a doctor to get it confirmed. Eczema is actually an allergic reaction. Babies who get it often develop allegies later. Newborns don't usually show signs of allegies at first, but since she's just a month old, it's likely that it just started. My son was that way, always had a rash sideburns, and a rash on his forehead. You may want to switch to a diet that eliminates a few of the more common allergins, like milk and milk products, wheat products, and all peanuts and treenuts for a week and see if it improves. Nightshade plants (potatoes, tomatoes and eggplants) are often responsible for allergies. If the elimination diet works and she looks better, then just introduce them back into your diet one at a time every three or four days. You might be able to find the culprit, or it may just dissappear, because her body may have a chance to adjust.