If you do it in a race, you hit "Save Replay" in the post-race menu. If it's just in a free run lobby, I don't think there is a way to save it unless you use a capture device other than the PS3.
AUP
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yes I wish there was a "start capture" button like in just cause 2. But alas as others have mentioned you can only save the replays of online races but not the free runs.
and CAM you must want to be a mod 
