I'm so embarassed, but I really don't see where you're going with this.

Here's what I wanted to do. Let $\displaystyle ABCD$ be the parallelogram in question, $\displaystyle \vec{a}=AB, \vec{b}=AD$. Surface area of that parallelogram is $\displaystyle S=|\vec{a}|h$, where $\displaystyle h$ denotes the height. That equals to $\displaystyle |\vec{a}| |\vec{b}|\sin \alpha$ which is $\displaystyle | \vec{a} \times \vec{b}|$.

So I was hoping I could express $\displaystyle \vec{a}, \vec{b}$ using $\displaystyle \vec{n}, \vec{m}$ and then calculate it using the above equation.

Does that make any sense?