Determinant – Find the corner product of two multivariate vectors and derive the meaning

I was trying to solve:
$$ -2dx_ {1} land dx_ {4} (
begin {bmatrix} 2 \ 3 \ -5 \ 2 end {bmatrix}, begin {bmatrix} 2 \ 3 \ 4 \ -5 end {bmatrix} $$

I solved this problem by removing the scalar and then finding the determinant of the matrix given by (let $ u $ to be the first column vector and $ v $ to be the following):
$$ det begin {bmatrix} -2 * dx_1 (u) & -2 * dx_1 (v) \ dx_4 (u) & dx_4 (v) end {bmatrix} = – 2 * det begin {bmatrix} dx_1 (u) & dx_1 (v) \ dx_4 (u) & dx_4 (v) end {bmatrix} = – 2 * det begin {bmatrix} 2 & 2 \ 2 & -5 end {bmatrix} = – 2 (-14) = 28 $$
We must then explain what this number represents and that is where my confusion influences. I understand that this represents the surface of the parallelogram formed from $ u $ and $ v $ being projected on the $ partial_ {x_1} partial_ {x_4} $ plane. Should this area be positive or negative? Without the scalar -2 $ the oriented area would go back since $ u $ at $ v $ would be clockwise, but with the scalar, it would be in the opposite direction of the hands of a watch not to change the sign? I guess my main confusion comes from whether or not the determinant has integrated orientation and that the scalar of one of the forms-one affects the orientation?