Yes. There was a time when the moon rotated on its axis faster than it travels around the earth. That would cause the moon to show all of its surface to earth. But that didn't last long.
The moon was so close to earth when it formed that it was lower than the geostationary satellites’ orbit around our planet. It's possible that it was as close as just 25 thousand kilometers from the surface. At that short distance, the tidal forces are huge, so earth’s gravity rapidly slowed down the moon's rotation. That happened so fast that just some 100 million years after it formed, the moon was already tidally locked to earth. In other words, the moon was already showing just one side to the earth just 100 million years after its birth.
But the fact is that the moon does show a little more than half of its surface. Here’s a sequence of photos taken by James McCarthy of the full moons that took place during almost one year:
You can see how the moon wobbles, showing and hiding parts of its surface, which allows us to actually see 59% of the illuminated surface from earth.
That's called libration and is caused by a couple of factors, but none of them related to the moon's rotation. That is, the moon is presently in perfect tidal lock with earth, and if it wasn’t for the other factors (orbit inclination, eccentricity and the observer's relative position change as the earth rotates), we would always see the same 50% of the moon's surface.