I need to ask a question about vectors and cross product? When you take the determinant on 3 vectors, you calculate and get the volume of that specific shape, correct?
When you take the cross-product of 2 vectors, you calculate and get the area of that shape and you also get the vector perpendicular to the plane, correct?

Kind Of.
When you take the determinant of a set of vectors, you get the volume bounded by the vectors.
For instance, the determinant of the identity matrix (which can be considered as a set of vectors) gives the volume of the solid box in $n$ dimensions. A $3\times3$ identity matrix gives the area of a cube.
However, when you calculate cross products, the matrix of whose determinant you take has the first row to be the the unit vectors in the $n$ dimensions.
For instance
\begin{align}
\det\begin{pmatrix}
\hat{i}&\hat{j}&\hat{k}\\1&0&0\\0&1&0
\end{pmatrix}=1 \hat{k}
\end{align}
It does NOT return a scalar value.
