How to plot a vector field with arrow length and background color for strength and direction

I want to visualize a 2D vector field, e.g.,

Table(Exp(-(x^2 + y^2)/4) {y, -x}, {x, -3, 3, 0.2}, {y, -3, 3, 0.2})

in a way where the arrow length denotes the vector length and the background color indicates the local vector orientation. Just something like the following. How can one achieve this in Mathematica?enter image description here