Computing optimal transport maps between high-dimensional and continuous distributions is a challenging problem in optimal transport (OT). Generative adversarial networks (GANs) are powerful generative models which have been successfully applied to learn maps across high-dimensional domains. However, little is known about the nature of the map learned with a GAN objective. To address this problem, we propose a generative adversarial model in which the discriminator's objective is the $2$-Wasserstein metric. We show that during training, our generator follows the $W_2$-geodesic between the initial and the target distributions. As a consequence, it reproduces an optimal map at the end of training. We validate our approach empirically in both low-dimensional and high-dimensional continuous settings, and show that it outperforms prior methods on image data.