This reference could be what you are looking for:
Digraph Laplacian and the Degree of Asymmetry:
We introduce a metric – the largest singular value $\lambda$ of $(\Gamma − \Gamma^T )/2$, where $\Gamma$ is the Laplacian of a directed graph – to quantify and measure the degree of asymmetry in the graph. The degree of asymmetry captures the overall "directedness" of the graph.
One could normalise $\lambda$ by the largest singular value of $\Gamma$ itself, for a "percentage" answer.