The attention mechanism can refine the extracted feature maps and boost the classification performance of the deep network, which has become an essential technique in computer vision and natural language processing. However, the memory and computational costs of the dot-product attention mechanism increase quadratically with the spatio-temporal size of the input. Such growth hinders the usage of attention mechanisms considerably in application scenarios with large-scale inputs. In this Letter, we propose a Linear Attention Mechanism (LAM) to address this issue, which is approximately equivalent to dot-product attention with computational efficiency. Such a design makes the incorporation between attention mechanisms and deep networks much more flexible and versatile. Based on the proposed LAM, we re-factor the skip connections in the raw U-Net and design a Multi-stage Attention ResU-Net (MAResU-Net) for semantic segmentation from fine-resolution remote sensing images. Experiments conducted on the Vaihingen dataset demonstrated the effectiveness and efficiency of our MAResU-Net. Our code is available at https://github.com/lironui/MAResU-Net.