Let us assume we have the $n$-dimensional sphere $\mathbb S^{n-1}$ and a matrix $A \in \mathbb R^{n \times n}$ such that every entry of $A$ is sampled from a normal distribution $\mathcal N (0,\sigma^2)$.
Let $x$ be a vector sampled uniformly at random from $\mathbb S^{n-1}$. What can be said about the distribution of $x^TA$?
What if we restrict the input space to the subset of $\mathbb S^{n-1}$ where every entry is non-negative?
Conditioned on the random matrix $A$, the random vector $x^TA$ is uniformly distributed on the ellipsoid $E_A := \{x \in \mathbb R^n \mid x^TBB^Tx = 1\}$, where $B=A^{-1}$. Thus, by the Bayes formula for total probability, the distribution of $x^TA$ is
$$ \begin{split} &\mathbb P(x^TA \in U) = \int \mathbb P(x^TA \in U \mid A)dp(A) = \int vol(U \cap E_A)dp(A),\\ &\text{for every Borel }U \subseteq \mathbb R^n. \end{split} $$
Unfortunately, I don't think things can be simplified any further...