Is it possible to expand the eye socket and bring my eye out further? (photo)

You need more detailed evaluation by an oculoplastic surgeon to determine what is possible. Technically, an orbital implant can be placed to push the eyeball forward but it may not possible in every patient as limited by orbit size. Ptosis surgery may be another treatment you can benefit from.

Thank you for your question. There are simple ways to improve the symmetry by opening the eyelid fissure via ptosis surgery.  A more involved surgery would be to place a bone graft or implant in the orbit to push the eye forward.  I suggest the less involving, eyelid surgery to achieve some symmetry .

Hello!
Thank you for the question and for the picture. 
It looks like you can benefit from a ptosis correction and epicanthoplasty in order to make your eye bigger. But depending on your condition, the surgery could help you or not, so it's better to see a plastic surgeon in person for a better consultation.
Good luck~