You invented snowboarding? - I called it "boogie-woogie one-ski." (Frostbite, 2005)
Can you ski on one ski? Chance are, you can — and you can't. Almost everyone can do some skiing on one ski, to the point where I usually incorporate it as a drill for first-day skiers. But very few of us can work our way down a mountain with only one ski under us. For most people, losing a ski at the top of a mountain means walking down to the bottom. For Pawel Babicki, it means finishing a race on one leg:
So, other than getting down after breaking a ski, why would you want to learn to ski on one ski? Skiing on a single ski requires finely tuned skills, because you can't rely on the other ski as a crutch. You have to be able to use one ski to do everything that you need, and not cheat by using the other ski to keep you in balance. This is no easy task, so I'm going to list a few drills in increasing difficulty, and you can judge your own ability to complete them.
The simplest task is to ski on one ski on very easy green terrain, down the fall line. I usually ask people to do this drill on their first day, even before we put a second ski on. If you get your balance too far back, you'll instinctively put your other foot down, so this is a good drill to ensure that your balance is roughly centered.
Next on the difficulty scale would be to traverse across the hill, and pick up your uphill ski. You should be able to do so without slipping sideways down the hill. Have a friend check the ski you lifted, and if you are in balance, the ski should be level with the ground. Again, this is a fairly easy drill that most beginners can accomplish, but having good fore-aft balance will help you succeed.
To raise the difficulty somewhat, try lifting the downhill ski in a traverse. Now we can determine if you are engaging both skis in a traverse, or only your downhill ski. Remember, the uphill edge of both skis should be engaged in a traverse, and any slipping here means that the uphill ski is not engaged. Try bending your uphill leg to get the ski engaged if you have any troubles here.
Next up you can attempt to turn with only one ski. Keep both skis on, and as you enter a turn lift the inside ski up off the snow. If your balance is good, this won't prove as difficult as it sounds. The reality is we normally get almost all of our turning force from the outside ski, so as long as you can handle the reduced based of support, you should be able to turn fine. Focus on fore-aft balance, and that will keep your lateral balance issues in check.
If most of the turning force comes from the outside ski, then you have probably realized that turning to the inside is where the difficulty lies. If you've never attempted this before, move to extremely easy terrain... The easiest green you have, preferably very short. Take your poles, but leave a ski behind. You'll want to just lift one foot, but it's a lot easier to ski on one ski without the extra weight dragging down your unused foot, so go ahead and leave it behind. Start slow, and ski down the slope.
The keys to accomplishing this are having your weight forward, and not letting the lack of a ski go to your head. Use pole touches, even on green terrain. Both legs should have a lot of flexion and extension, even the leg that never touches the snow. Extend it on the inside turn, and flex it on the outside turn. This will help you get the edge of the remaining ski fully engaged into the snow. Stay calm and confident, and do everything the same way you would if you were skiing perfectly on two skis. If you can use either ski to turn either direction, you're truly skiing efficiently!