My joints hurt and there is rough skin around my elbows. could this be arthritis?

Possibly: If the skin condition involving the elbows is psoriasis then there is a possibility of psoriatic arthritis, which occurs in 20% of the patients with psoriasis. This would generally be manifested as painful and swollen joints classically involving 2 or 3 joints.

Could be: You should see your internist for a serologic examination to rule out auto-immune conditions at your age. A dermatologist may also be of assistance in looking at your rough ski to determine if this is possible eczema or psoriatic arthritis.

Arthritis: This could be psoriatic arthritis. If this pain continues then a consult with a rheumatologist would be recommended.