Can Matrix Determinants Be Used to Find Optimal Area in Higher Dimensions?

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It is possible to find area of triangle or parallelogram in euclidean by using matrix determinant composed of unity, x coeffs and y coeffs in row1,2,3 respectively. Is it possible to do that in higher dimensions as well although it may be not as simple as in 2D case. In 3d matrix composed of x,y,z would instead give volume but I am looking for area of a plane suspended in space.
 
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vin300 said:
It is possible to find area of triangle or parallelogram in euclidean by using matrix determinant composed of unity, x coeffs and y coeffs in row1,2,3 respectively. Is it possible to do that in higher dimensions as well although it may be not as simple as in 2D case. In 3d matrix composed of x,y,z would instead give volume but I am looking for area of a plane suspended in space.
It's all explained here:

http://math.arizona.edu/~calc/Text/Section13.4.pdf
 
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I am thinking about a newer more interesting and seemingly difficult problem. The problem has a rectangular table with a triangular sheet to be laid upon, whose dimensions are greater than the table but the area is lesser. What would be the relative angle, and position of the triangle that gives optimum area on the field, minimum outside?