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:

1. h is a linear subspace

2. $exp(h)$ 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][1] 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.


  [1]: http://books.google.co.uk/books?id=TC7d3ZcqjfsC&lpg=PP1&dq=adams%20lectures%20lie%20groups&pg=PA17#v=onepage&q=&f=false