Regardless of when and where skiing actually began, in Mongolia or in Norway, the classic design and shape of modern skis began with Sondre Norheim in the 19th century. Prior to Sondre’s influence Norse skis were straight and simple, with a mere loop to hold feet to the ski. Norheim added shape and a binding that wrapped around the heel for more control. Below is a video showing how craftsmen made wood skis the old-fashioned way.
As to the origins of skiing, click here for a report confirming the influence of Norway on the roots of skiing.