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$SO(2n,\mathbb R)$ is the special orthogonal group of $\mathbb R^{2n}$, which is the group of $2n\times2n$ real orthogonal matrices with determinant 1.

If we take $U(n)$ the group of $n \times n$ unitary complex matrices, we can show that $U(n)$ can be injected in $SO(2n,\mathbb R)$. To the best of my knowledge, this injection is strict once $n$ is greater than 1. Therefore, $U(n)$ is not an adequate complex $n \times n$ matricial representation of $SO(2n,\mathbb R)$ when $n>1$. My question is, is there such representation and what is it?

The origin of this question comes when I tried to adapt the action of direct Euclidean isometries in even dimensions ($SO(2n,\mathbb R) \ltimes \mathbb R^{2n}$ ) on vertical vectors of $2n$ real entries, to an action on vertical vectors of $n$ complex entries.

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