Let $\mathfrak{g}$ be a Lie subalgebra of $\mathfrak{gl}(n, \mathbb{R})$ and $\exp(\mathfrak{g})$ its image by the exponential map. Although this set itself might not be a group, its (algebraic) closure $\langle \exp(\mathfrak{g}) \rangle$ forms a connected subgroup of $GL(n, \mathbb{R})$. Furthermore, taking the exponential map as a chart around identity, this group admits a compatible smooth structure so that inclusion in $GL(n, \mathbb{R})$ is an immersion. The result is really a connected Lie subgroup of the general linear group, and $\mathfrak{g}$ is its Lie algebra.
Is it true that every connected subgroup of $GL(n, \mathbb{R})$ arises in this way? What results would I need to show this?