Showing 4 planes in the 4D space are in a same 3D hyperplane

Showing 4 planes in the 4D space are in a same 3D hyperplane

Given four 2D planes in the 4D space, how can I specify if they lie on a
3D hyperplane or not. Assume $\langle x_i, y_i, z_i, w_i \rangle$ for
$i=1,\cdots,4$ be the normal vectors of the planes. Is that enough to show
that they are not linearly independent, i.e., the following determinant is
zero?
$$ \begin{vmatrix} x_1& y_1& z_1& w_1 \\ x_2& y_2& z_2& w_2 \\ x_3& y_3&
z_3& w_3 \\ x_4& y_4& z_4& w_4 \end{vmatrix} $$
What about more than 4 dimensions? If we want to show that in $n$
dimensional space, $n$ hyperplanes of dimension $n-2$ lie on a hyperplane
of dimension $n-1$, is that enough to show that the normal vectors of the
planes are not linearly independent?