Hmm, if I have a browser open to YT and it usually does automatically select the highest resolution based on screen ability and if the browser is full screen or not. If this isn't happening for the OP then something is wrong. Even when I click the full-screen option from the video bar at the bottom it will change the resolution accordingly.
For instance, my 1920x1200 monitor has a browser open that uses ~80% of the screen width so the resolution in a YT video will be 720P. If I hit the full screen mode it automatically goes to 1080P.