Geometric understanding of the Cross Product

Say you have vectors $v$ and $w$. Let there cross product be denoted by $x$ so that:

$$v \times w = x$$

According to Wikipedia:

$$x_x = v_yw_z - v_zw_y$$ $$x_y = v_zw_x - v_xw_z$$ $$x_z = v_xw_y - v_yw_x$$

This is equivalent to saying:

$$x_x = \left|\begin{matrix}v_y&v_z\\w_y&w_z\end{matrix}\right|$$ $$x_y = \left|\begin{matrix}v_z&v_x\\w_z&w_x\end{matrix}\right|$$ $$x_z = \left|\begin{matrix}v_x&v_y\\w_x&w_y\end{matrix}\right|$$

The determinant can be interpreted as the area spanned by the column vectors. Could you give me a geometric explanation of why the area of those above vectors give the coordinates of the cross product? Thanks!

EDIT: An interpretation for a 2x2 matrix is fine too

Let $P$ be the paralellogram spanned by the vectors $a$ and $b$ and let $\phi$ be the angle between $a$ and $b$. Then we have \begin{eqnarray*} (Area(P))^2 & = & |a|^2|b|^2(\sin(\phi))^2 \\ & = & |a|^2|b|^2-|a|^2|b|^2(\cos(\phi))^2 \\ & = &|a|^2|b|^2-(a\cdot b)^2 \\ & = & (a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)-(a_1b_1+a_2b_2+a_3b_3)^2\\ & = &(a_2b_3-a_3b_2)^2+(a_3b_1-a_1b_3)^2+(a_1b_2-a_2b_1)^.2 \end{eqnarray*} So the area of $P$ is equal to the length of the vector $(a_2b_3-a_3b_2,a_3b_1-a_1b_3,a_1b_2-a_2b_1) = a \times b$. With the coordinates you gave.

There is a geometric interpretation of this in terms of the areas of the projections of the parallelogram $P$ spanned by $\bf{v}$ and $\bf{w}$ to the coordinate planes. Let $\bf{n}$ be a unit vector perpendicular to the plane spanned by $\bf{v}$ and $\bf{w}$. Projecting this plane to the $yz$-plane is a map which decreases area by a factor of $\bf{i}\cdot \bf{n}=\bf{n}_x$, where $\bf{i}=(1,0,0)$. The two planes intersect in a line, for which projection preserves length, and for a line orthogonal to this line of intersection, the projection multiplies length by $\bf{n}_x$, so area changes by a factor of $\bf{n}_x$. As you note, each coordinate of the cross product is the (signed) area of the parallelogram spanned by the projections of $\bf{v}$ and $\bf{w}$ to each coordinate plane (which are the column vectors you refer to). So the length of the cross product is $|\bf{n}|\cdot Area(P) = Area(P)$. So this derives this formula (modulo signs) from the property that the cross product $\bf{v}\times \bf{w}$ is perpendicular to $\bf{v}$ and $\bf{w}$, and has magnitude $|\bf{v}\times \bf{w}|=Area(P)$, up to choices of signs. The sign choices are forced by the choice of dot product, which by changing the coefficients could be of different signatures.