Area of triangle in $\mathbb R^3$ given $3$ coordinates

The geometric interpretation of the magnitude of the cross product $\left\|\vec v \times \vec w\right\|$ is the area of the parallellogram spanned by the vectors $\vec v$ and $\vec w$ and the triangle with vertices $\vec o$, $\vec v$ and $\vec w$ is exactly half of that parallellogram, hence its area is given by $\tfrac{1}{2}\left\|\vec v \times \vec w\right\|$.

This interpretation can be clear from the definition (depending on how you define the cross product), or it follows from the formula $\left\|\vec v \times \vec w\right\|=\left\|v\right\|\left\|v\right\|\sin\theta$ ("base times height") where $\theta$ is the angle between $\vec v$ and $\vec w$.