Is there a way to rotate a plane/face in a way that its normal will face 3d cursor?
Not directly as far as I know, but you can use a constraint to your advantage as a workaround.
Add a Track To constraint to a plane, set it up to track any empty object. Make it point in Z direction and use Y Up.
You can now move this Empty object freely through your scene, and your plane will always face it. If you duplicate the plane, any clones should keep the behavior in sync.
While you can't make the plane face the cursor directly, you can now select the empty, and using the Shift + S Snap menu move the empty to the cursor easily with the option Snap > Selection to Cursor, cirtually making them point to the current cursor position.