Talk to Pediatrician on Skin Allergy

hello doctors. my son is 42 days old. He started having some red pimples on his face and back of his neck when he was 3 weeks old. He caught some infection in his tummy. The doctor prescribed antibiotic Cefim. I gave him 2ml twice a day for 5 days. The doctor said that my son might get some rashes on his skin during recovery. Now, he has red pimples on cheeks , back of his neck, forehead and some on his chest. The pimples red and without any pus. they are dry and feel like dry hard patches when i touch. His eyelid is also affected. The condition sometimes seems better but sometimes worse. I just clean his face with wet cotton and applied petroleum jelly. What else can i do?