This is somewhat of a conceptual question that I need resolved as I may have interpreted things wrong:

When we have two vectors, U and V, taking their cross products (UxV) produces a resultant vector that is orthogonal to both U and V.

Now if we consider a 3 by 3 matrix in which each row would represent a vector, respectively U, V, W. If we compute the wronskian of the matrix is the resulting vector orthogonal to U, V and W?

If no could you please clarify why and is there a way to compute a cross product of 3 or more vectors? (possibly by introducing higher dimensions)

Thank you for the help.