I use Shadowplay and Twitch but you can use OBS.
You can either stream it and in my twitch settings set to save my past broadcast for 14 days. Then I click on the past broadcast video on my twich videos page and it gives you a highlight option on the bottom right. It bring you to an cool little editor and you can make a quick highlight. (all video files saved to twitch servers)
If you dont feel like streaming you can always set shadow play to save the last 5 or 10 mins of video by using alt F10. This will create a video file on your hard drive which you can play with any video player