The crash was caused by part of a tree falling on the car. A few moments before the crash we see the camera focus on the part of the tree that is breaking. This caused the car to skid off on the ice and through the side of a bridge, it was the fall from the bridge that killed the First Lady.
Quote from wikipedia:
Former Special Forces U.S. Army Ranger Mike Banning (Butler) is the lead Secret Service agent assigned to head the Presidential Detail. He maintains a personal, friendly relationship with President Benjamin Asher (Eckhart), First Lady Margaret (Judd) and, especially, their son Connor (Jacobsen). During a Christmas evening drive from Camp David to a campaign fundraiser, a tree branch falls and strikes the front of the president’s convoy, making the vehicles skid out of control on black ice on a bridge. The lead vehicle crashes through the guard rail and falls into the icy river below, leaving the presidential limousine teetering on the edge of the bridge. Banning is able to save President Asher, but Margaret and two other agents die when their vehicle falls and crashes.