Aside from any particular book, I'd say that you need a human being reviewing and giving feedback on your proofs. This is a type of writing for consumption by other people. One of the main things is that a proof should be clear, explanatory, and insightful. Partly this criteria depends on the level of expertise of the expected audience.
Now, I'm not entirely sure what is the best way to arrange this outside of class. Get a "proof pen pal" with whom you swap proofs and give feedback for how clear it was? An online study-group type situation? Post attempted proofs to SE Mathematics and ask for feedback/improvements? (Kind of like how SE Software Engineering, I think, accepts code-review posts).
When you do that, bear in mind that's how actual professional articles get written. Someone writes a draft of one or more sections, they get sent around to colleagues for commentary, and the draft gets re-written and re-written until everyone gives the thumbs-up to it. So if you can get that process going, then you'll have an additional leg up on the people-skills that a math researcher needs.