The area of a parallelogram formed by two vectors is \(\text{base } \times \text{ height}\). We can represent the base as the magnitude of one of the vectors and the height as the magnitude of the perpendicular component of the other vector:
This means:
$$\text{Area }= \Vert\textbf{u}\Vert \ \Vert\textbf{v}\Vert \ \sin(\theta) $$
The above is just another way to represent the cross product:
$$\begin{gather} \Vert \textbf{u} \times \textbf{v} \Vert = \Vert\textbf{u}\Vert \ \Vert\textbf{v}\Vert \ \sin(\theta) \\ \text{Area }= \Vert \textbf{u} \times \textbf{v} \Vert \end{gather}$$