I think jump seating to Reagan International would be a fun trip. You'll most likely be able to experience the visual approach down the Potomac, and if you have a lay over for a few hours you could go see Arlington National Cemetery, or walk downtown and see the monuments.
I did an internship with Independence Air in DC in the spring of 05 and I never got tired traveling downtown, even though it took me a few times to get the streets down
Good Luck on your internship!