Is it worth doing skin tests for allergies, on a 3 month old baby?

No. The child should be at least 3 years old before allery testing is performed.
Minimal value. Allergies to airborn substances may develop over a lifetime but the immature kid system doesn't yield dependable results until 4 or so. Food items can generate reproducable results in infants but i rarely test until 1y & for a kid having chronic sinusitis/otitis. Dietary exclusion is a better test for food issues for young infants.Most congestion in 3m is irritants in the air.
Allergy testing. . Generally speaking, skin testing can be performed even in infancy, and as young as one month of age. However, the skin of very young children may not be as reactive as older children and adults, and therefore the results need to be interpreted more carefully.