The Schrödinger bridge is a stochastic process that finds the most likely coupling of two measures with respect to Brownian motion, and is equivalent to the popular entropically regularized optimal transport problem. Motivated by recent applications of the Schrödinger bridge to trajectory reconstruction problems, we study the problem of sampling from a Schrödinger bridge in high dimensions. We assume sample access to the marginals of the Schrödinger bridge process and prove that the natural plug-in sampler achieves a fast statistical rate of estimation for the population bridge in terms of relative entropy. This sampling procedure is given by computing the entropic OT plan between samples from each marginal, and joining a draw from this plan with a Brownian bridge. We apply this result to construct a new and computationally feasible estimator that yields improved rates for entropic optimal transport map estimation.