Non-symmmetric generalization of Gaussian integral.

Question

Suppose that $A$ is a symmetric, positive definite $n \times n$ matrix. Then we have the relation \begin{align*} \int_{- \infty}^{\infty} \exp( - \frac{1}{2} A_{ij} x_j x_i) dx_1 \dots d x_n =  \sqrt{ \text{Det} (2\pi A^{-1})} \end{align*} Does this formula have an extension if $A$ is not symmetric?