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**Drexel28** Perhaps a more conceptual way of looking at it, if that kind of thing makes you happy, is that if $\displaystyle V,W$ are isomorphic vector spaces then $\displaystyle \text{GL}(V),\text{GL}(W)$ are isomorphic groups. Now, evidently $\displaystyle \dim_\mathbb{R}\mathbb{C}^n=2n$ so that $\displaystyle \mathbb{C}^n\cong\mathbb{R}^{2n}$ as real vector spaces, and so $\displaystyle \text{GL}_n(\mathbb{C})\cong \text{GL}(\mathbb{C}^n)\cong\text{GL}(\mathbb{R}^{ 2n})\cong\text{GL}_{2n}(\mathbb{R})$.