What kind of testing is done to determine milk allergy?

How does one know if the baby is lactose-intolerant or has cow's milk allergy ( ie allery to cow's milk drunk by a feeding mom / that in infant formula ) ? I heard about tests to determine this. Does anyone know what kind of tests determine lactose intolerance vs cow's milk allergy ? My baby used to be very fussy. The fussiness has reduced since I stopped drinking cow's milk ( I feed ). I do however eat yoghurt. So cow's milk was a problem. Just dont know if the problem was the lactose in the cow's milk or the protein in the cow's milk or anything else about the cow's milk. The reason I'm asking this ques is that I plan to start supplementing with formula. I am hesitant to use soy-based formula because of what I've read up about side-effects of soy-based formula on infants.