Motivated by the widespread use of large gridded data sets in the atmospheric sciences, we propose a new model for extremes of areal data that is inspired by the simultaneous autoregressive (SAR) model in classical spatial statistics. Our extreme SAR model extends recent work on transformed-linear operations applied to regularly varying random vectors, and is unique among extremes models in being directly analogous to a classical linear model. An additional appeal is its simplicity; given a proximity matrixW, spatial dependence is described by a single parameter rho. We develop an estimation method that minimizes the discrepancy between the tail pairwise dependence matrix (TPDM) for the fitted model and the estimated TPDM. Applying this method to simulated data demonstrates that it is able to produce good estimates of extremal spatial dependence even in the case of model misspecification, and additionally produces reasonable estimates of uncertainty. We also apply the method to gridded precipitation observations for a study region over northeast Colorado, and find that a single-parameter extreme SAR model paired with a neighborhood structure which accounts for longer range dependence effectively models spatial dependence in these data.