Skeleton-based human action recognition has received widespread attention in recent years due to its diverse range of application scenarios. Due to the different sources of human skeletons, skeleton data naturally exhibit heterogeneity. The previous works, however, overlook the heterogeneity of human skeletons and solely construct models tailored for homogeneous skeletons. This work addresses the challenge of heterogeneous skeleton-based action representation learning, specifically focusing on processing skeleton data that varies in joint dimensions and topological structures. The proposed framework comprises two primary components: heterogeneous skeleton processing and unified representation learning. The former first converts two-dimensional skeleton data into three-dimensional skeleton via an auxiliary network, and then constructs a prompted unified skeleton using skeleton-specific prompts. We also design an additional modality named semantic motion encoding to harness the semantic information within skeletons. The latter module learns a unified action representation using a shared backbone network that processes different heterogeneous skeletons. Extensive experiments on the NTU-60, NTU-120, and PKU-MMD II datasets demonstrate the effectiveness of our method in various tasks of action understanding. Our approach can be applied to action recognition in robots with different humanoid structures.