Brown's essay has an open-ended topic, and you can, if I am not mistaken, select to do a topic of your choice for the Common App essay. So yes, you could do that.
My D did. It is perfectly OK, as long as you are not using a specific essay topic from one school/application to apply to another. As long as you are writing an open-ended topic, you can submit it anywhere.
For Brown, I submitted an essay that was almost 800 words. I did pare it down to a version that's about 500 words... which one should I do for the Common App? Shall I go for conciseness, or slightly more personal-ness?
it says that they want to tell them something that will give them "a sense of who you are, how you think, and what issues and ideas interest you most" . What if my essay does not address the issues and ideas that interest me most?