"We can check both span and linear independent in one time by finding det, can't we? "

Depends what you mean by check span.

To check if vectors are linearly independent you can make a matrix and calculate the determinant. If the determinant is non-zero then there is a unique solution to the homogeneous equation (trivial solution) and therefore the vectors are linearly independent. Note that you can only do this if the matrix made of the vectors is square. Otherwise use Gaussian elimination.

Now if you want to know if a vector is in the span of some set of vectors, you might be able to form a square augmented matrix and take the determinant of that. If that determinant is 0 then you have infinite solutions and the vectors are linear combinations of each other.

If you want to check if 2 vectors span R^2, then you make sure they are linearly independent. Once again, you can test this with the determinant. It all comes down to linear independence and how you want to use it.