First in $\mathbb{R}^3$ for $u=(u_1,u_2,u_3), \ v=(v_1,v_2,v_3), \ w=(w_1,w_2,w_3)$ the triple product gives :
$\det(u,v,w)=(u \times v).w$, where $\times$ is the vectorial product.
Now for $\mathbb{R}^n$ and $u_1=(u_{1_1},...,u_{1_n}),\ ...\ ,u_n=(u_{n_1},...,u_{n_n})$ do we have a formula of the form :
$\det(u_1,...,u_n)=(u_1\times...\times u_{n-1}).u_n$ ?
It seems to only work for an orthonormal basis.
Thanks in advance !