$\{T(a)| \;T:\mathbb{C}^n\longrightarrow\mathbb{C}^n$ is a $\mathbb{C}$-linear isometry with $\det(T)=1\}=\mathbb{S}^{2n-1}$

Question

Let $a,b\in\mathbb{C}^n$ ($a\neq b$ and $|| a||=||b||$). How can one prove that there is a $\mathbb{C}$-linear isometry $T:\mathbb{C}^n\longrightarrow\mathbb{C}^n$ (that is $T(z)=U\cdot z$ where $U$ is an unitary matrix) such that $T(a)=b$ with $\det(U)=1$.

If $T$ is a reflection such that $T(a)=b$ hold, but $\det(T)=-1$.

$||a||=\sqrt{|a_1|^2+\ldots+|a_n|^2}$

Any help would be appreciated.