  We derive bulk asymptotics of skew-orthogonal polynomials (sop)
$\pi^{\bt}_{m}$, $\beta=1$, 4, defined w.r.t. the weight $\exp(-2NV(x))$, $V
(x)=gx^4/4+tx^2/2$, $g>0$ and $t<0$. We assume that as $m,N \to\infty$ there
exists an $\epsilon > 0$, such that $\epsilon\leq (m/N)\leq \lambda_{\rm
cr}-\epsilon$, where $\lambda_{\rm cr}$ is the critical value which separates
sop with two cuts from those with one cut. Simultaneously we derive asymptotics
for the recursive coefficients of skew-orthogonal polynomials. The proof is
based on obtaining a finite term recursion relation between sop and orthogonal
polynomials (op) and using asymptotic results of op derived in \cite{bleher}.
Finally, we apply these asymptotic results of sop and their recursion
coefficients in the generalized Christoffel-Darboux formula (GCD) \cite{ghosh3}
to obtain level densities and sine-kernels in the bulk of the spectrum for
orthogonal and symplectic ensembles of random matrices.
