Skull stripping magnetic resonance images (MRI) of the human brain is an important process in many image processing techniques, such as automatic segmentation of brain structures. Numerous methods have been developed to perform this task, however, they often fail in the presence of neuropathology and can be inconsistent in defining the boundary of the brain mask. Here, we propose a novel approach to skull strip T1-weighted images in a robust and efficient manner, aiming to consistently segment the outer surface of the brain, including the sulcal cerebrospinal fluid (CSF), while excluding the full extent of the subarachnoid space and meninges. We train a modified version of the U-net on silver-standard ground truth data using a novel loss function based on the signed-distance transform (SDT). We validate our model both qualitatively and quantitatively using held-out data from the training dataset, as well as an independent external dataset. The brain masks used for evaluation partially or fully include the subarachnoid space, which may introduce bias into the comparison; nonetheless, our model demonstrates strong performance on the held-out test data, achieving a consistent mean Dice similarity coefficient (DSC) of 0.964$\pm$0.006 and an average symmetric surface distance (ASSD) of 1.4mm$\pm$0.2mm. Performance on the external dataset is comparable, with a DSC of 0.958$\pm$0.006 and an ASSD of 1.7$\pm$0.2mm. Our method achieves performance comparable to or better than existing state-of-the-art methods for brain extraction, particularly in its highly consistent preservation of the brain's outer surface. The method is publicly available on GitHub.