Hi M.,
How often does he get this rash? Sounds like it comes and goes? If so, I doubt he's allergic to milk. He probably eats some diary every day, so the rash would be there all the time.
One of my sons got this type of rash on his face when he was little. It was a milk allergy. He didn't react to the formula he had been drinking, but did react when we switched him to whole milk. It did not come and go though. It got progressively worse for two months until I switched him back to formula. Then it went away... and came back when I tried giving him whole milk again.
If your son's rash comes and goes, then (if it is a food allergy) it is probably something he only eats occasionally. Something like eggs or shrimp perhaps? I get eczema from egg. For me, the rash flares up (gets redder and itchier) the day after I eat eggs. Then it takes several days (4-5) of not eating it for the rash to go away.
The 8 most common food allergy triggers are: peanuts, tree nuts, dairy, soy, egg, wheat, fish, and shellfish. Any time his rash flares up, try to think back to what he ate that day and the day before and watch for a pattern.
Food allergies are a bear to identify through trial and error. I would recommend allergy testing instead. Your son is old enough that he should be able to deal with it. (Our son was a toddler when he was tested. Both the skin-prick and blood draw were really tough on him.)
Good luck!!
-D.