  We present a new explicit formula for the determinant that contains
superexponentially fewer terms than the usual Leibniz formula. As an immediate
corollary of our formula, we show that the tensor rank of the $n \times n$
determinant tensor is no larger than the $n$-th Bell number, which is much
smaller than the previously best-known upper bounds when $n \geq 4$. Over
fields of non-zero characteristic we obtain even tighter upper bounds, and we
also slightly improve the known lower bounds. In particular, we show that the
$4 \times 4$ determinant over $\mathbb{F}_2$ has tensor rank exactly equal to
$12$. Our results also improve upon the best known upper bound for the Waring
rank of the determinant when $n \geq 17$, and lead to a new family of
axis-aligned polytopes that tile $\mathbb{R}^n$.
