The student pilot reported that prior to takeoff, he was aware of a left crosswind of about 5 knots and compensated by applying extra left aileron. During the takeoff roll at about 45 to 50 knots, the airplane started to veer to the left. The pilot stated that the airplane “felt like it had lifted off the ground” in a gradual left roll and he was unable to correct with right aileron or sufficient rudder. Once the airplane departed the runway, he reduced the power and applied the brakes, but the airplane skidded down an embankment and collided with a hill. The impact separated the engine from the mounts and the nose landing gear was sheared off, which caused substantial damage to the fuselage. The pilot reported that he likely had more left aileron than was necessary for the crosswind at the beginning of the takeoff roll, and he did not reduce the control application as the airspeed increased. The pilot reported that there were no mechanical malfunctions or failures of the airplane that would have precluded normal operation.