Is there a notation for the "magnitude with sign" of the cross product?

Question

If $a,b\in\mathbb{R}^3$, then $$a\times b = \hat{n}\|a\|\|b\|\sin\alpha$$ where $\hat{n}$ is perpendicular to $a,b$ and $\alpha$ is the angle between $a$ and $b$. Does there exist a standard notation for the "signed norm" $$\|a\|\|b\|\sin\alpha$$?