Distribution of $aXa^T$ for normal distributed vector $a$ Let $a$ be $1\times n$ random vector with entries chosen independently from normal distribution with zero mean and unit variance. What is the distribution of $aXa^T$ for a given $n\times n$ matrix $X$. 
If $X$ is symmetric matrix, then the above is a Wishart distribution. What is $X$ is not symmetric?

$X$ can be written as a sum $X_{s} + X_a$, where $X_{s}$ is symmetric and $X_a$ is antisymmetric.  But $a X_a a^\text{T} = 0$ for any $a$ and any antisymmetric $X_a$, so WLOG, suppose $X = X_s$.  Then your penultimate sentence applies.
