Let $\Omega \subset \mathbb C^n$ have $C^2$ boundary near $p \in \partial\Omega$. If $\partial\Omega$ is strictly pseudoconvex at $p$, then there are local holomorphic coordinates $w=(w_1,\dots,w_n)$ centred at $p$ and a local defining function $r$ for $\Omega$ such that \begin{align*} r(w)=2\operatorname{Re}(w_n)+\sum_{j=1}^{n-1}|w_j|^2+o(|w|^2) \end{align*} as $w \to 0$, after multiplying $r$ by a positive scalar. Conversely, if a boundary point admits such a holomorphic coordinate normal form with a positive definite Hermitian quadratic part in the $w_1,\dots,w_{n-1}$ variables, then $\partial\Omega$ is strictly pseudoconvex at that point.