For three arbitrary points, $A$, $B$, $C$, with position vectors $\bar a, \bar b, \bar c$, the equation of the plane can be expressed as:
$$(\bar a \times \bar b) \hat x + (\bar b \times \bar c) \hat y + (\bar c \times \bar a) \hat z = \bar a \cdot \bar b \times \bar c$$
I understand where the formula comes from, and how to derive it, but what is the intuitive meaning behind having the gradient of (each basis of) the plane as the magnitude of vector of points crossed (or the exterior product) ?