The answer is "it depends."
It depends on what type of food sensitivity, if any, is involved.
Milk allergy is caused by IgE antibodies to one or more protein components of the milk. Symptoms can include cough, wheezing, nausea, headache, hives and eczema. This type of hypersensitivity can be tested by either skin prick or blood test (RAST), so a dairy elimination diet may not be necessary for testing purposes. (Reference)
Your son may have a delayed-type hypersensitivity to dairy (mediated by T-lymphocytes, not IgE). There are a few delayed skin tests for antigens such as tetanus, yeast, tuberculosis and poison ivy (Ref.), but I don't know if a delayed cow milk skin test is available. Usually the best way to test for delayed food sensitivities is to completely eliminate the suspected food (dairy) from the diet for a few weeks. If the symptom (eczema) improves, then re-challenge with dairy to see if the rash worsens. Partial elimination of dairy may work for this test, but complete elimination would probably work better. (Ref.)
Regarding treatment, delayed hypersensitivities are dose-dependent, while IgE-mediated hypersensitivities are not. This means that only complete elimination of dairy will be effective if your son has IgE-mediated milk allergy, but a partial dairy elimination may be effective for a delayed cow milk sensitivity.