Emotion recognition is an important component of affective computing, and also human-machine interaction. Unimodal emotion recognition is convenient, but the accuracy may not be high enough; on the contrary, multi-modal emotion recognition may be more accurate, but it also increases the complexity and cost of the data collection system. This paper considers cross-modal emotion recognition, i.e., using both electroencephalography (EEG) and eye movement in training, but only EEG or eye movement in test. We propose cross-modal contrastive representation distillation (CMCRD), which uses a pre-trained eye movement classification model to assist the training of an EEG classification model, improving feature extraction from EEG signals, or vice versa. During test, only EEG signals (or eye movement signals) are acquired, eliminating the need for multi-modal data. CMCRD not only improves the emotion recognition accuracy, but also makes the system more simplified and practical. Experiments using three different neural network architectures on three multi-modal emotion recognition datasets demonstrated the effectiveness of CMCRD. Compared with the EEG-only model, it improved the average classification accuracy by about 6.2%.