Do I need to take an antibiotic to treat my eczema?

No. This can be due to an allergic reaction to food hypersensitivity or a reaction to flour, yeast or grain itch mite. It could also be due to a parasite infestation or herpes virus. Most usually it is due to a combination of genetic susceptibility and environmental factors, with a breakdown of the immune system at the point of the skin barrier.
No. Eczema itself is not an infection. It it usually caused by allergies. The best approach is allergy therapy that can treat all types of exposures: foods, chemicals, and inhalants (dust, mold, etc.). In the meantime, allergy meds can sometimes be helpful.