Is every unitary operator a self-adjoint one?

Question

Let $X$ be a finite dimensional inner product space over $\mathbb{C}$. Let $T:X\to X$ be a linear map. Then is the following true?

1. $T$ is unitary implies $T$ is self-adjoint

I think this is correct. $T$ is unitary implies that $T^{\theta} T=TT^{\theta}=1$. This means that $\langle T^{\theta}T(x), y\rangle=\langle x, y\rangle$ where $T^{\theta}$ is complex conjuate of the matrix of $T$. This further means that $\langle T(x), y\rangle=\langle x, T(y)\rangle$, i.e. $T$ is self-adjoint.

Have I done it correctly or I am missing something?