I would most definitely not send a letter of interest. You applied to their school and just went there for an interview...obviously you are interested. Not to say that you didn't, but your opportunity to show your interest by showing genuine excitement and knowledge about GW's program during your interview. During my interview at GW, both my interviewers allowed me the chance to ask about anything...so you had the opportunity. IMO (feel free to disagree anyone), sending a letter so soon after your interview would perhaps show impatience, something they are not looking for. And if you got an interview, they are most likely not too concerned with any update. You've made it past the hard part in getting an interview. The interview means they like who you are on paper and want to make sure you're a normal human being. The only reason you would need to send an update at this point would be if you got wait-listed later on, but aside from curing cancer, no update will have much of an impact on their decision.
Send a thank you email, thank them for the opportunity to interview, perhaps mention one part of the school/interview you enjoyed, and express how you are looking forward from hearing back from then.
Take a deep breath, have a beer (or two or three) and enjoy the moment. They said 8-10 weeks and I heard back in under 5, so you might not have to wait as long as they say.
Good luck.