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The determinant of the matrix of its vectors gives the measure of an $n$-dimensional parallelogram.

For example, in $2$ dimensions, the area spanned by vectors $v$ and $w$ is \begin{array}{|cc|} v_1 & w_1 \\ v_2 & w_2 \\ \end{array} and so forth for a $3$ or more -dimensional parallelogram.

How is possible to visualize that, or understand that intuitively?

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    Have a look at "Linear Algebra via Exterior Products" by S. Winitzki: http://sites.google.com/site/winitzki/linalg , pagg.30-31.2012-04-30

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