In the vector space $\mathbb R^2$, writing $\phi(v,w)$ for the angle from vector $v$ to vector $w$, we have $$ \sin\phi(v,w)=\frac{\det(v,w)}{|v||w|}. $$ More generally, in $\mathbb R^n$, we could define $$ S(v_0,\dots,v_{n-1})=\frac{\det(v_0,\dots,v_{n-1})}{|v_0|\cdots|v_{n-1}|}, $$ and consider $S$ as generalizing $\sin\circ\phi$ to $n$-dimension. $S$ has the following property: (i) when some vectors are along the same direction, then the parallelepiped formed by the normalized vectors has no volume, and $S(v_0,\dots,v_{n-1})=0$. (ii) when they are orthogonal to each other, then $S$ achieves the maximum value 1. $S$ has other properties, such as being alternating.
Is there an analogue for cosine? If we call it $C$, then for $\mathbb R^2$, we should have $$ C(v,w)=\frac{\langle v,w\rangle}{|v||w|}. $$ $C$ probably should have the following property: (i) when all vectors are along the same direction, it achieves the maximum value 1. (ii) when some vectors are orthogonal to each other, $C(v_0,\dots,v_{n-1})=0$.
The question has been asked before, but has got no answer so far. An alternative formulation might also have some value.
SSD (https://math.stackexchange.com/users/228795/ssd), Wedge Product Formula For Sine. Analogous Formula Generalizing Cosine to Higher Dimensions?, URL (version: 2015-04-09): Wedge Product Formula For Sine. Analogous Formula Generalizing Cosine to Higher Dimensions?