  We study the fluctuations of eigenvalues from a class of Wigner random
matrices that generalize the Gaussian orthogonal ensemble. We begin by
considering an $n \times n$ matrix from the Gaussian orthogonal ensemble (GOE)
or Gaussian symplectic ensemble (GSE) and let $x_k$ denote eigenvalue number
$k$. Under the condition that both $k$ and $n-k$ tend to infinity with $n$, we
show that $x_k$ is normally distributed in the limit. We also consider the
joint limit distribution of $m$ eigenvalues from the GOE or GSE with similar
conditions on the indices. The result is an $m$-dimensional normal
distribution. Using a recent universality result by Tao and Vu, we extend our
results to a class of Wigner real symmetric matrices with non-Gaussian entries
that have an exponentially decaying distribution and whose first four moments
match the Gaussian moments.
