possible milk allergy?

My baby has patches of dry skin and when I got him out of the bath they were really red. He coughs a lot and sneezes a lot, been constipated for 2 days, been in a new formula for two weeks, been hotter than normal but no fever, really red around just his butthole even before constipation, wondering if this sounds like a milk allergy and if the rash looks like eczema or an allergic reaction