Yep, snow happens when the temperature at the cloud layer is low enough. Ground temp can be higher and the snow will still make it to the ground. Might melt immediately, or if it's snowing enough it could form an insulating layer that keeps snow on the ground a bit longer.

Honestly, snow and air temperature have a bit of a counterintuitive relationship. It can also be too cold to snow, so a lot of really cold places are actually really dry and get very little precipitation because of that.