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.

Related Questions

I am 25 and my joints hurt and there is rough skin around my elbows. Could this be signs of arthritis?

Possible not likely. But psoriatic arthritis is a rare possibility that should be evaluated by your physician. Far more likely causes are eczema and/or traumatic osteoarthritis.
Psoriatic arthritis. This could be psoriatic arthritis. You should consult with a rheumatologist for an accurate diagnosis and proper treatment regimen.

My joints are painful and there is rough skin around my elbows. Is it possible I have arthritis at 25?

Yes. If your joints hurt, if is more that six weeks and you have inflammation, redness, stiffnes or increase in size then you need to see a rheumatologist. The rough skin in your elbows can be dryness or any skin disease. It has to be evaluated.
Yes. Rough skin - sounds like psoriasis. See pits in any fingernails -- probably psoriasis. There is something called psoriatic arthritis --go see a specialist.