Fix $v \in \mathbb R^n$, the probability matrix $A$ with normally distributed entries has an eigenbasis that $v$ projects nontrivially is $1$

Question

Suppose $v \in \mathbb R^n$ is a fixed nonzero vector and $A \in M_n(\mathbb R)$ is a random matrix where each entry is taken from a standard normal distribution over $\mathbb R$. We know that the probability $A$ is diagonalizable is $1$. My question is: if we concern those matrices that are diagonalizable (thus with an eigenbasis), what is the probability that those with an eigenbasis and $v$ projects nontrivially on each eigenvector in the eigenbasis, i.e., the probability measure of the set \begin{align*} \{A \in M_n(\mathbb R): & \text{each entry is generated from a standard normal,}\\ & \text{if $A=S\Lambda S^{-1}$ (spectral decomposition) then $Sv$ has no zero component}\}. \end{align*}