Hi, I have learned a lot about allergies in the last 3 years. Both my sons (2.5 yr.old and 9 months) have food and environmental allergies. If your baby has had eczema since he was very young it is very likely allergies are involved. If you nursed, foods you ingested could have been the culprit and just as likely, allergies to formula. Indoor and outdoor allergies can be big factors as well: animal dander, molds, dust/dust mites, pollen, detergents, household cleaners etc....
My suggestion would be go for the allergy testing. You will receive enormous amounts of information that can help you deal with your baby's everyday symptoms and hopefully get to the root cause of the allergies. Both my sons were tested before 1 year. It is an intense process but allergies affect quality of life in many ways and knowing what the causes are can really improve how your child feels each day.
In the meantime, you can take some preventative measures: air purifiers with HEPA filters make a big difference (place in the baby's room and play areas), make sure animals are not in any sleeping areas, use the A/C in spring/fall/summer during high pollen and mold count days b/c it filters the air, encase your child's bedding with allergy covers, clean relentlessly w/o your child in the same room your cleaning, check out Little Noses new "allergy block" - little tube of non-medicinal ointment to block allergens around nose, Saline also by Little noses works wonders to flush out allergens (non-medicated, safe to use all day as much as needed).
You can also talk to your doctor about using Zyrtec or Claritin children's medicine. If you decide to go the allergist route they will be able to help you choose the best medicine to use and will give you information about non medical interventions to help prevent and alleviate symptoms.
If you are near the Richmond, VA area, I'll be glad to refer you to the allergy practice my boys see. Feel free to contact me directly ____@____.com
I wish you and your baby well!