We prove a uniqueness theorem for solutions to the wave map equation in the natural class, namely (u, ∂tu) ∈ C([0, T); Ḣd/2) × C1([0, T); Ḣd/2-1) in dimension d ≥ 4. This is achieved by estimating the difference of two solutions at a lower regularity level. In order to reduce to the Coulomb gauge, one has to localize the gauge change in suitable cones, as well as to estimate the difference between the frames and connections associated with each solution and to take advantage of the assumption that the target manifold has bounded curvature.