The pilot was attempting to land at the destination airport after a cross-country flight at night. During landing, the airplane touched down, and when the pilot applied the brakes, the airplane began to skid. The airplane then departed the left side of the runway, and as the pilot attempted to regain directional control of the airplane, it veered back onto the runway off the right side. With about 75 feet of the 3,000-foot-long runway remaining, the pilot elected to turn the airplane left in an attempt to reverse its direction. Before the turn could be completed, the airplane struck the airport perimeter fence, resulting in substantial damage to both wings. The pilot reported no mechanical malfunctions or failures of the airplane that would have precluded normal operation, and a postaccident examination of the airplane revealed no anomalies with either the flight controls or the brakes.