Does downhill skiing improve your running...?

Just to be specific, I am speaking about downhill and not X-country skiing and by running I mean trail / long distance running.

The experiences I am drawing on are from our recent trip to Snowmass, Colorado.
Most of the skiing was done between 3000m to 4000m asl. For perspective, Australia's highest peak is 2,228m or 7,310ft.

The Oxygen levels are obviously less at this altitude, but did it make a difference to my training...?
Because I am not a performance athlete, I can say that I do not believe that it made a material difference, at least none to speak of. I do believe that if I was training for 1 month or more at this altitude that when I returned to sea level, my bodies ability to use Oxygen more efficiently would be improved.

What I do believe has made a difference to my running fitness level is the constant use of my quads whilst skiing.  So, skiing has increased my leg muscles ability to store glycogen and this is evident now when I am running hills.