EDDITTT: In the first paragraph I write (d,c), in the second paragraph (c,d) which are different. These are often called ordered or oriented bases.
Even in $\mathbb R^2$ this does not have a single answer. Given the way you chose the letters, the order is critical: if $(d,c)$ is a rotation and scaling of $(a,b),$ then your expression is indeed $0.$
However, if $(c,d)$ is a rotation and scaling of $(a,b),$ you get anything between certain bounds. For example, let $a,d$ point in the same direction, so that $d$ is a positive multiple of $a.$ Then $c$ is a negative multiple of $b.$ In symbols, some $\lambda > 0,$ you get $d = \lambda a, \; \; c = - \lambda b.$ So
$$ b \cdot c - a \cdot d = - \lambda b^2 - \lambda a^2 = -2 \lambda a^2, $$
where $a^2 = b^2$ is the squared length of either vector.
Take the previous paragraph and simply negate $c,d,$ now you get $2 \lambda a^2$
The key word here is ORIENTATION.