Why do I get Lie(matrix Lie group)=$\mathfrak{gl}_n(\mathbb{R})$?

Question

There is something I don't understand about Matrix Lie groups. They are defined as closed subgroups of $GL_n(\mathbb{R})$ but we can show that a closed subgroup of a topological group is opened as well. So if $H$ is a Lie matrix group, $H$ is opened in $Gl_n(\mathbb{R})$ so the tangent space of $H$ and $GL_n(\mathbb{R})$ are isomorphic so their Lie algebra are the same (i.e. $\mathfrak{gl}_n(\mathbb{R})$. But it seems this is not true because if we take $H=SO(n)$, it's lie algebra is formed by antisymmetric matrices and not all the matrices). How to correct this reasoning?

Answer 1

It is not true that every closed subgroup of $GL_n(\Bbb R)$ is also open. For instance, $\{\operatorname{Id}_n\}$ is a closed subgroup which is not open.