One vector spans a line. Two linearly independent vectors span a plane. And $\geq 3$ linearly independent vectors span a hyper-plane. Therefore, we need to figure out how many linearly independent vectors are in this set.
The determinant will only tell you whether your set of vectors as a whole are linearly independent. If your determinant turns out to be zero, as it did in this case, then we cannot deduce how many independent vectors are in the set - only that the number is less than $3$. So at this point, all we know is that $Span(u, v, w)$ will turn out to be either a line or a plane.
Now, an easy way to proceed from here is to plug your vectors into a matrix as its rows. Then, perform Gauss-Jordan elimination to get the matrix into row-echelon form. From here, the number of nonzero rows in row echelon form will give the dimension of the row space of the original matrix. That is, it will give the number of linearly independent row vectors in the original matrix.