Isomorphism between $O(2n,\mathbb{R})$ and $O(n,n,\mathbb{R})$ and same question for their Lie algbera

Question

Is there any isomorphism of Lie groups between $O(2n,\mathbb{R})$ and $G:=\{ X \in M_{2n\times 2n}(\mathbb{R}) \mid X^tSX = S\} $ where $S$=$\begin{pmatrix} & {I_n}\\ I_n & \end{pmatrix}$. If not , are their Lie algebras isomorphic ? What is an example of an isomorphism(s) ,if any?