OK, let's see ...
180 LEDs at 6 mA each, notionally 10.8 A but a 10 A supply will probably be fine.
I honestly cannot recall exactly what 20 awg wire is but the calculator notes it as half a square millimetre in cross section so it should be good for a meter or two at 10 Amps.
Note you will need to connect the power across to the LED strip at least every metre along, including both ends if you intend to operate it at anything near full brightness white.
But that is just the power wiring, which does not go through the Arduino, so you should probably find some lighter gauge wire to use to connect to it. Note my explanation of how to wire it in this post.