Let $A$ be an $n\times n$ gaussian matrix whose entries are i.i.d. copies of a gaussian variable, and $\left\{ a_{j}\right\} _{j=1}^{n}$ be the column vectors of $A$. How to show that the probability $\mathbb{P}\left(d\geq t\right)\leq Ce^{-ct}$ for some $c,C>0$ and every $t>0$, where $d$ is the distance between $a_{1}$ and the $n-1$-dimensional subspace spanned by $a_{2},\cdots,a_{n}$.

Thanks a lot!