Well, the crux of the matter is to cook up the linear subspace of $gl(n)$ that will eventually be the subalgebra.
If $H$ is the subgroup: set $h={X\in gl(n): (\forall t \in \mathbb{R}) \, exp(tX) \in H }$.
You need to show two not completely obvious things:
$h$ is a linear subspace
$exp(h)$ is a nbhd of $1$ in $H$ (give $H$ the induced topology from $GL_n(\mathbb{R})$).
Adams has ingenious short arguments for these on pages 17-19 of his Lectures on Lie groups which require nothing but a little real analysis.
I remark that all this uses no manifold or Lie theory beyond the necessary definitions and the argument works completely without change for closed subgroups of any Lie group.
