Using the two way distance, we introduce the concepts of weak metric dimension of a strongly connected digraph $Γ$. We first establish lower and upper bounds for the number of arcs in $Γ$ by using the diameter and weak metric dimension of $Γ$, and characterize all digraphs attaining the lower or upper bound. Then we study a digraph with weak metric dimension $1$ and classify all vertex-transitive digraphs having weak metric dimension $1$. Finally, all digraphs of order $n$ with weak metric dimension $n-1$ or $n-2$ are determined.