Can you hear your exact voice like others hear yours?

No, not always.

Some people lack the ability to hear themselves the way other people hear them. For example, look at singing competitions on television. Some people think they sound like amazing singers, but in reality they sound like a screeching cat having his tail stamped on.

A way to hear yourself properly is to record your own voice and listen to it, rather than relying on how your brain thinks you sound.