See, for example, <A HREF="https://arxiv.org/abs/0801.1730">Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices</A> (2008). In particular the large-$n$ result

$$\text{Prob}(E_{\rm smallest}\geq x)\simeq\exp\left[-N^2\Phi\left(\frac{x+\sqrt{2n}}{\sqrt{n}}\right)\right],\;\;-\sqrt{2n}<x<0$$