Are you locking your elbows (don't do that!) Is your bike too big and the reach to the handlebars too long? How's your core strength?

If you haven't had your bike properly fitted to you, I highly recommend it. It'll cost about $100 but it's well worth it.

If your bike is properly fitted to you, then focus on keeping your elbows slightly bent and your shoulders relaxed.

If you have a weak core, you'll end up supporting yourself on your hands. Start strengthening your core (I like Pilates best) so that you can support yourself better.