In my opinion, the most skill is needed in the hand that has the most difficulties.
For me, that IS the bowing hand.

Work on playing with a controlled but completely relaxed wrist. Pretend you're drawing lines with that hand. This is what gives you those really fast triplets, snaps and other quick direction changes. You do use your whole arm, but the really small, fast movements should come from the wrist

Where the bow is on the strings and angle will change the sound. You can practice in a mirror