In my experience, it has been pretty easy to ask why they don't want you. Some larger companies have policies about not telling you, but any small company probably won't mind answering a polite email with a reasonably straightforward answer. And in larger companies, you should be asking each individual interviewer for a business card to follow up with a thank you email, so you could reach out to them directly; they aren't as sensitive to the HR policies.

You get an email saying "We're sorry..." and you just reply "Thank you for the opportunity. If you don't mind, I'd appreciate any information you are willing to share about your decision. I'm always looking to improve myself and rely on your honesty to identify my grow areas. Thanks!"

What's the worse that happens? They reject you again by not giving you a straight answer? The whole point of learning to handle rejection is to take advantage of the fact that much of the time you can win just by showing up to the game.