Multimodal emotion recognition (MER) aims to identify emotions by utilizing affective information from multiple modalities. Due to the inherent disparities among these heterogeneous modalities, there is a large modality gap in their representations, leading to the challenge of fusing multiple modalities for MER. To address this issue, this work proposes a novel attention-based MER framework by integrating representation subspace mapping with unimodal auxiliary loss for enhancing multimodal fusion capabilities. Initially, a representation subspace mapping module is proposed to map each modality into two distinct subspaces. One is modality-public, enabling the acquisition of common representations and reducing the discrepancies across modalities. The other is modality-unique, retaining the unique characteristics of each modality while eliminating redundant inter-modal attributes. Then, a cross-modality attention is leveraged to bridge the modality gap in unique representations and facilitate modality adaptation. Additionally, our method designs an unimodal auxiliary loss to remove the noise unrelated to emotion classification, resulting in robust and meaningful representations for MER. Comprehensive experiments are conducted on the IEMOCAP and MSP-Improv datasets, and experiment results show that our method achieves superior performance to state-of-the-art MER methods. Keywords: Multimodal emotion recognition, representation subspace mapping, cross-modality attention, unimodal auxiliary loss, fusion