Eigenvalue difference of $2 \times 2$ real symmetric random matrix

Question

Let $H:= \begin{pmatrix} h_{11} & h_{12} \\ h_{12} & h_{22} \end{pmatrix} $ be a $2 \times 2$ real symmteric random matrix with continuously distributed independent entries and real eigenvalues $\lambda_1$ and $\lambda_2$. I want to show that $P(|\lambda_1-\lambda_2| \le \epsilon) \sim \epsilon^2 $, in the sense that the probability scales like $\epsilon^2$ for small $\epsilon$. My idea was first to compute the eigenvalue difference in terms of the matrix elements, which results, if I'm not mistaken, in $|\lambda_1-\lambda_2|=\sqrt{(h_{11}-h_{22})^2+4h_{12}^2}$, so one should be able to argue that this difference is $0$ iff $h_{11}-h_{22}$ and $h_{12}$ both are $0$. Does this observation help to obtain the desired scaling property? If yes, how can I proceed from here?