The flight instructor and a private pilot were practicing takeoffs and landings on runway 12 (2,800 feet x 50 feet). The wind was from the southwest at 6 to 10 knots, resulting in a right crosswind. Although the runway was dry, there were layers of snow on both sides, 6 to 12 inches deep. After making different types of takeoffs and landings, a soft field takeoff was attempted. The brakes were held and full power was applied. After brake release, the pilot brought the nose up to try and lift off in ground effect. The high nose attitude prevented either pilot from seeing straight down the runway and right aileron was applied to correct for the right crosswind. The airplane lifted off the ground and drifted rapidly towards the left side of the runway. The instructor said he didn't realize how close they were to the runway edge. The pilot said he attempted to correct, but the left wheel contacted the snow bank and he reduced the power to idle. The airplane skidded unevenly in the snow and nosed over. Both wing upper skins and wing tips were buckled, the right lift strut and the lower aft empennage were bent.