Are there any common patterns for making a 6x6 (or bigger) LEGO board have a ability to rotate using technic with gears and such? Has to be without the help of the existing 2x2 lego part that rotates, its not big enough.
Seen from above:
Then you can attach the 6x6 plate on top.
There's actually a specialized part to accomplish this task. Take a look at the Technic Turntable (2855):
This part can be used to create sturdy rotating platforms that can be controlled by gears. It consists of two parts (2855 and 2856) that snap together to allow for smooth rotation about the center. Here's an example from the classic Mobile Crane set:
With the crane boom mounted:
Note that this turntable is out of production, so it may be easier to find the new studless version, though it will obviously be a little more difficult to attach regular bricks to this one. The studless version is part 48452 and 48168. Here's what is looks like with the two halves snapped together: