Oh Carol, take me! Pat and I have been to Big Sky a couple of times. We did some backcountry skiing in the NW corner of Yellowstone, right where Rte 191 enters the park. There's a trailhead there, and you can take off in any direction - steep or flat, whatever you like. Wide open spaces and herds of elk on the far ridges.
Lone Mountain Ranch is a beautifully laid out and groomed XC center that we took advantage of a couple of times, close to the village about 6 miles from the alpine area (which had the best skiing I have ever experienced).
Now you've done it...I have to go back soon.
West Yellowstone (sounds a bit crowded) has all kinds of winter activity going on and many ski trails for all levels.
My son and daughter-in-law opted one day for the classic guided tourist venue and came back with awesome pictures of bison, waterfalls, bald eagles, hot springs, and more.