Depends on how skinny you are, a lot of people think they are skinny but are not really at least to me. For your height I'd say less than 117 lbs could be considered skinny, I'm 5.7 and weight about that and I've been skinny all my life. When I was at college, summer used be a problem for me, being self conscious about my weight. Shorts definitely look weird if you have long legs
Seriously though, it's just a question of accepting your body.