If you don’t have or can’t afford an external monitor, there is a simple hack. Put your laptop vertically like I have in the photo. Please note, laptops have few moving parts inside, like your hard drive (no problem with SSDs, since they are solid state) and cooling fans. Generally if you be gentle, they shouldn’t be affected, but please take extra care while moving the laptop. Most laptops (except some new gaming laptops) have most of their ports on the sides, so keep in mind the ports on the side on which the laptop stand become temporarily unusable. So you need to see which side you have least/avoidable ports and stand on that side.
Once done, change the display orientation to portrait or portrait (flipped) depending on the direction of your rotation. Then you can play the game and record.