You don't need to move your arms much to play the piano well. Just watch a video of Vladimir Horowitz to see that. He hardly moved his hands up at all most of the time, but he had phenomenal control over the keyboard. Not only that, he was famous for having great rapport with his audiences.
It is generally good to have relaxed and free motion with your whole body when playing any instrument. When I was beginning at the piano, one teacher gave me an exercise for freeing up my motion:
- Raise your hand up to shoulder level or higher. Let your hand fall onto a C major triad (just C, E, G, no octave C). Don't let your fingers or wrist collapse, but allow them to be flexible and absorb some of the energy from the impact.
- Once your hand lands on the chord, move your wrist in little circles. Allow your fingertips to roll around on the keys, but make sure the chord sustains (without pedal).
I found the second part of this exercise to be more helpful than the first.
Note: this is not how you should play loudly. When you want to play a chord very loudly, you should play "up", as other people have commented. Consider the recording of Horowitz playing Chopin's A flat major Polonaise. Starting at 1:11, consider how he plays that top chord (around 1:18). He doesn't drop his hand down, he starts with his hand on the key, raises it up a tiiiny bit before he plays, and it shoots up after he plays. This is very typical of piano playing.
Another idea for practice: play a one-hand piece/exercise. Use your other hand to conduct or just gesture with the music. Even something as mechanical as a Hanon exercise will have something to gesture about if you listen for it.
Another idea for practice: BREATHE!!! Music is attached to your breath. When you're playing with both hands, show the musical gestures with your breath. Inhale before playing a big note, especially if it isn't on the beat.
That's all I can think of for now, I hope this helps.