Sending a LOI probably has some value, but I think you should rethink what you'd be doing in that letter. You don't "ask" for admission and you don't even bring up scholarships, especially before you're admitted. They know almost any student would attend if given a full tuition scholarship (unless they got a comparable offer at a similar school). Instead, you want to provide detailed and insightful reasons why WashU is your top choice and why you'd fit in well. There are lots of articles on how to write LOI's on SDN and elsewhere, I'd start there.