There is no balanced round-robin that you could use to do this. For example I can make the following schedule:
[(B E):(C D)]
[(C E):(D F)]
[(B D):(A C)]
[(E F):(A B)]
[(A F):(B C)]
[(B F):(D E)]
[(A E):(C F)]
But, A never partners D, and also A & D have one game fewer than the others. The opponent balance is poor, 3 pairs only oppose once, while the pair E & F oppose 3 times.