Proximity search on heterogeneous graphs aims to measure the proximity between two nodes on a graph w.r.t. some semantic relation for ranking. Pioneer work often tries to measure such proximity by paths connecting the two nodes. However, paths as linear sequences have limited expressiveness for the complex network connections. In this paper, we explore a more expressive DAG (directed acyclic graph) data structure for modeling the connections between two nodes. Particularly, we are interested in learning a representation for the DAGs to encode the proximity between two nodes. We face two challenges to use DAGs, including how to efficiently generate DAGs and how to effectively learn DAG embedding for proximity search. We find distance-awareness as important for proximity search and the key to solve the above challenges. Thus we develop a novel Distance-aware DAG Embedding (D2AGE) model. We evaluate D2AGE on three benchmark data sets with six semantic relations, and we show that D2AGE outperforms the state-of-the-art baselines. We release the code on https://github.com/shuaiOKshuai.