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        "quote": "Note that these matrices form a subspace of $\\mathbf{C}^{n^2}$ of half the dimension, so of real dimension $n^2$. $\\mathfrak{u}(n)$ is a real vector space of dimension $n^2$, but it is NOT a space of real $n$ by $n$ matrices. It is the space of [<u class=\"concept-term\" data-concept-reference=\"concepts/linear-algebra/skew-adjoint-matrix\" data-concept-identity=\"urn:uuid:fef8514e-8fc7-492e-b6ff-ed8d15f79734\" data-concept-route=\"/concept-identities#concept-fef8514e-8fc7-492e-b6ff-ed8d15f79734\" id=\"first-dc893b45ec0132267c33-en\">skew-Hermitian matrices</u>](../../../../../concepts/linear-algebra/skew-adjoint-matrix), which are generally complex. While the matrices are complex, only real linear combinations of skew-Hermitian matrices are skew-Hermitian (recall that multiplication by $i$ changes a skew-Hermitian matrix into a Hermitian matrix). Within this space of skew-Hermitian complex matrices, if one looks at the subspace of real matrices one gets the sub-Lie algebra $\\mathfrak{so}(n)$ of antisymmetric matrices (the Lie algebra of $SO(n)\\subset U(n)$).",
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