The morphology of retinal vessels is closely associated with many kinds of ophthalmic diseases. Although huge progress in retinal vessel segmentation has been achieved with the advancement of deep learning, some challenging issues remain. For example, vessels can be disturbed or covered by other components presented in the retina (such as optic disc or lesions). Moreover, some thin vessels are also easily missed by current methods. In addition, existing fundus image datasets are generally tiny, due to the difficulty of vessel labeling. In this work, a new network called SkelCon is proposed to deal with these problems by introducing skeletal prior and contrastive loss. A skeleton fitting module is developed to preserve the morphology of the vessels and improve the completeness and continuity of thin vessels. A contrastive loss is employed to enhance the discrimination between vessels and background. In addition, a new data augmentation method is proposed to enrich the training samples and improve the robustness of the proposed model. Extensive validations were performed on several popular datasets (DRIVE, STARE, CHASE, and HRF), recently developed datasets (UoA-DR, IOSTAR, and RC-SLO), and some challenging clinical images (from RFMiD and JSIEC39 datasets). In addition, some specially designed metrics for vessel segmentation, including connectivity, overlapping area, consistency of vessel length, revised sensitivity, specificity, and accuracy were used for quantitative evaluation. The experimental results show that, the proposed model achieves state-of-the-art performance and significantly outperforms compared methods when extracting thin vessels in the regions of lesions or optic disc. Source code is available at https://www.github.com/tyb311/SkelCon.