On Wikipedia here, the following is asserted.
The unitary group is the $3$-fold intersection of the orthogonal, symplectic, and complex groups:$$\text{U}(n) = \text{O}(2n) \cap \text{Sp}(2n, \mathbb{R}) \cap \text{GL}(n, \mathbb{C}).$$
What is the quickest way to see this?
The unitary group is the group of all $G \in GL_n(\mathbb{C})$ preserving the standard inner product $\langle \cdot, \cdot \rangle : \mathbb{C}^n \times \mathbb{C}^n \to \mathbb{C}^n$. The standard inner product on $\mathbb{C}^n$ has a real and imaginary part, and is preserved iff its real and imaginary parts are preserved. After splitting up each coordinate of $\mathbb{C}^n$ into its real and imaginary parts, you can compute that the real part is the standard inner product on $\mathbb{R}^{2n}$, and the imaginary part is $i$ times the standard symplectic form on $\mathbb{R}^{2n}$ (maybe up to a sign or something).