The problem of improving image quality in ultrafast ultrasound (US) imaging by means of regularized iterative algorithms has raised a vast interest in the US community. These approaches usually rely on standard image processing priors, such as wavelet sparsity, which are of limited efficacy in the context of US imaging. Moreover, the high computational complexity of iterative approaches make them difficult to deploy in real-time applications. We propose an approach which relies on a convolutional neural network trained exclusively on a simulated dataset for the purpose of improving images reconstructed from a single plane wave (PW) insonification. We provide extensive results on numerical and in vivo data from the plane wave imaging challenge (PICMUS). We show that the proposed approach can be applied in real-time settings, with an increase in contrast-to-noise ratio of more than 8.4 dB and an improvement of the lateral resolution by at least 25 %.