Someone already mentioned determinants.  Here is a related $n$-ary operation, the vector product in dimension $n+1$:  Fix a basis $b_1,\dots,b_{n+1}$ of $\mathbb R^{n+1}$.
To $n$ elements $v_1,\dots,v_n$ of $\mathbb R^{n+1}$ assign
the unique vector $v_{n+1}$ that is orthogonal to $v_1,\dots,v_n$, such that
$v_1,\dots,v_{n+1}$ is of the same orientation as $b_1,\dots,b_{n+1}$ and such that the
length of $v_{n+1}$ is the $n$-dimensional volume of the parallelopiped spanned by
$v_1,\dots,v_n$.