I can understand your concerns,I think you need to confirm from your doctor that which type of eczema
are you having ? thats important!
Now comming towards the treatment plan,i would recommend following:
1: If the antibiotics are not working then you need to discuss with your doctor about sending a wound swab to the lab so that you can find it out to which antibiotics you are sensitive and resistant.
2:Emolients are as much important as the anti biotics are,so discuss the proper one for you with the dermatologist
3: avoid yourself from irritants like harsh and perfumed soaps,creams and other cosmetics ad even frequent washing with water.
4: use a soap free face cleanser.
i hope you ll find the discussion helpful.