Given an affine coordinate system $[O; e_{1,2,3}]$ and its associated Gram matrix $G$ (i.e. $G=(e_i,e_j)_{ij}$ where the parenthese denote inner product), is there any slick way to retrieve the normal vector of the plane $$A_1x_1+A_2x_2+A_3x_3=0.$$
If we can find two independent vectors that span the plane, we're basically done. But there are infinitely many of them and they aren't that straightforward to compute.
Sure. Calculate $n=G^{-1}A$, where $A$ is the vector with coordinates $A_i$. Then note that the dot product of $n$ and $x$ is given by $$n^\top Gx = (G^{-1}A)^\top Gx = A^\top G^{-1}Gx = A^\top x = 0.$$