3 is old enough to get her into a "Ski Wee" program on the mountain. No time like the present to get the little one excited about skiing. The programs for 3-6 yo's focus on fun and confidence more than skills - often with play zones on the hill, sleigh rides to/from the lodge, etc. Usually plenty of breaks for hot chocolate, potty and warm-ups. Sometimes they offer both indoor and outdoor play. It can be pricey, but well worth the cost to get you time on the slopes while introducing her to skiing and playing in the snow.
As long as she's the least bit adventurous, she'll love it. When my kids were little, we would take 'em along and put them in to a 1/2 day "Ski Wee" session. They're in a new area, seeing new and exciting things and having some fun on the snow. My wife would only ski 1/2 day and she was fine with baby sitting in the pm (she would read, my son would nap after a busy morning on the mountain). I would probably start with a 1/2 day in the program and see if she's up for the full day. Even then, you'll have to take time to stop in every couple of hours to check on her and make sure everything's OK.
Skiing with the family means adjusting your expectations. If your goal is to go shred the gnar from first chair to last, it's not a family trip .. go by yourself. But if you're willing to adjust your expectations to include spending time with the family on the slopes, you and your wife can still have plenty of time to explore the mountain while devoting some quality time to your daughter and introducing her to a great family sport.
You also need to set the expectations for your daughter before you go. Get brochures and pictures from the mountain. Tell her about the cool instructors, the fun zones, sleigh rides or whatever else the mountain has and get her excited about going.
As much as we all like to ski, make sure you have alternatives as well. Dog sled or sleigh rides, pool or water park, a short snowshoe or hike. Resorts with an active ski town usually have plenty of alternatives to a day on the slopes.
3 is on the young side, so her capacity for multiple days of Ski Wee may be limited, but there's nothing that says you can't have a great time with the family on a super winter vacation .. where you get to go skiing as well.