In this paper, a novel contingency-aware method for N-2 security-constrained transmission expansion planning is proposed. To ensure that the transmission construction plan satisfies the N-2 security criterion, the proposed method takes advantage of the adjustable robust optimization (ARO) framework and upgrades it. We construct a discrete uncertainty set in which component failures are treated as uncertain events that are handled as binary variables. In addition to the failure of existing lines and generators, we explicitly model the failure of candidate lines. The proposed model comprises a master problem that makes the transmission construction decision, and a series of subproblems that can detect not only the worst contingency, but also the potential contingencies. Computational studies on the IEEE RTS 24-bus and IEEE 118-bus test systems are carried out to validate the effectiveness of the proposed method. Compared with the deterministic method and ARO method in the literature, the proposed method has higher computational efficiency.