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Tumor segmentation using magnetic resonance imaging (MRI) plays a significant role in assisting brain tumor diagnosis and treatment. Recently, U-Net architecture with its variants have become prevalent in the field of brain tumor segmentation. However, the existing U-Net models mainly exploit coarse first-order features for tumor segmentation, and they seldom consider the more powerful second-order statistics of deep features. Therefore, in this work, we aim to explore the effectiveness of second-order statistical features for brain tumor segmentation application, and further propose a novel second-order residual brain tumor segmentation network, ie, SoResU-Net. SoResU-Net utilizes a number of second-order modules to replace the original skip connection operations, thus augmenting the series of transformation operations and increasing the non-linearity of the segmentation network. Extensive experimental results on the BraTS 2018 and BraTS 2019 datasets demonstrate that SoResUNet outperforms its baseline, especially on core tumor and enhancing tumor segmentation, illuminating the effectiveness of second-order statistical features for the brain tumor segmentation application.