When figure skating really started to be recognized as a sport was in the 1960s. In this picture to the left, we see figure skater Peggy Fleming who won the 1968 Olympic figure skating competition. During the 60s it was very common for a costume to be one piece, resembling a unitard with a skirt attached to the bottom. Resembling a unitard, most costumes were made with spandex material that was stretchy.
http://figureskating.about.com/od/whattowear/ss/History-Of-Figure-Skating-Clothes.htm#step-heading