Given $n-1$ orthogonal unit length vectors $b_1, b_2, \ldots, b_{n-1}$ in the vector space $\mathbb{C}^n$, it's clear that there exsits a unique $b_n \in \mathbb{C}^n$ with unit length, which is orthogonal to all $b_1, \ldots, b_{n-1}$. In other words, $b_1, \ldots, b_{n-1}$ can be uniquely extended to an orthonormal basis.

Is there a simple explicit formula for $b_n$ in terms of $b_1, \ldots, b_{n-1}$?