The directed network embedding problem is to represent the nodes in a given directed network as embeddings (i.e., low-dimensional vectors) that preserve the asymmetric relationships between nodes. While a number of approaches have been developed for this problem, we point out that existing approaches commonly face difficulties in accurately preserving asymmetric proximities between nodes in a sparse network containing a large number of low out- and in-degree nodes. In this paper, we focus on addressing this intrinsic difficulty caused by the lack of information. We first introduce the concept of virtual negative edges (VNEs), which represent latent negative relationships between nodes. Based on the concept, we propose a novel DIrected NE approach with VIrtual Negative Edges, named as DIVINE. DIVINE carefully decides the number and locations of VNEs to be added to the input network. Once VNEs are added, DIVINE learns embeddings by exploiting both the signs and directions of edges. Our extensive experiments on four real-world directed networks demonstrate that adding VNEs alleviates the lack of information about low-degree nodes, thereby enabling DIVINE to yield high-quality embeddings that accurately capture asymmetric proximities between nodes. Specifically, the embeddings obtained by DIVINE lead to up to 10.16% more accurate link prediction, compared to those obtained by state-of-the-art competitors.