My 6 month old started antibiotics with penacilen and he had the runs and his ezcma has flared up is it an allergic reaction?

Maybe: Allergic reactions to medicines come in many shapes and sizes, but loose stools are a very common side effect, not allergy, of many oral antibiotics. Eczema flares can be a symptom of allergy to many things, including drugs. Discuss with your doctor if the benefits of continuing the current antibiotic outweigh the potential downsides of stopping it due to your child's eczema..

No: It is unlikely for mild diarrhea and an eczema flare are due to penicillin allergy.