So you have the pin on the arduino uno which is ground, are you able to use a jumper wire to send the ground source out the other end of the wire to the alligator clip for the antistatic wrist strap?
I was about to write that I wouldn't do that, but in view of the post I linked to - I am changing my mind.
The point to the wrist strap it to have you, and the thing you are touching, at the same potential (whatever that is relative to the known Universe). Now if you connected the strap to the earth on your power point, that would not achieve anything unless your Arduino was also grounded (eg. via a USB cable).
So using the ground wire on the Arduino sounds sensible enough. Personally I don't usually do that, however if I have moved around (and might have shuffled over some carpet, etc.) I always touch some metal part to ground myself (for example, a metal switch on a lab power supply) and discharge any static that might have built up by moving around on the floor.