Is it possible that I have a broken bone? Just went airborne on my bike and fell on outstretched arm.

Certainly possible. Severe pain after such a fall in the wrist or forearm certainly could justify an x-ray to rule out that possibility. Also persistence of significant pain for more than a few days would warrant x-ray to exclude a broken bone as apposed to sprain.
Yes. Falling on an outstretched limb can easily cause a fracture. You should have it checked by a physician.