Almost, but not quite. You could say "n! is the product of the natural numbers <= n." By this definition 0! is an empty product because there are no natural numbers <= 0.
However, this same reasoning would say (-1)! = 1, (-2)! = 1, etc. And while you could define negative factorials that way, it's better in practice to leave negative factorials undefined.
However, this same reasoning would say (-1)! = 1, (-2)! = 1, etc. And while you could define negative factorials that way, it's better in practice to leave negative factorials undefined.