The Cessna 170B "started to swerve to the right" after the pilot corrected for a slight left drift while attempting a cross wind landing. Despite an attempt by both the pilot and flight instructor to arrest the right turn by applying "all controls and left brake," the airplane continued to veer to the right and skidded off the runway. During the skid the left wing contacted the runway and was substantially damaged. The instructor reported that the wind was a direct crosswind from the right at approximately 5 knots. The winds recorded at the nearest airport, 30 miles to the east, were from an easterly direction at approximately 6 to 9 knots. In addition, the pilot checked no mechanical malfunctions/failures on the National Transportation Safety Board form 6120.1 he submitted.