As you all know, there is geometric place of points of cross of two planes (given as plane vectors) explicitly written simply as $\mathbb{r}=c_1\mathbb{n_1}+c_2\mathbb{n_2}+\lambda\mathbb{n_1}\times\mathbb{n_2}$
More general is $x^i = c_1 n_1^i + c_2 n_2^i + \lambda \epsilon_{ijk} n_1^j n_2^k$, where i,j,k denotes dimension, {x,y,z} for $R^3$, $\epsilon$ is standard form of cross product of vectors, which is actually antisymmetric operator, rotating variables.
But, in more then 3 dimensional space, there is no rotational operator acting $R^3 \rightarrow R^3$. Instead, absolutely antisymmetric operator acts like cyclic index rotation group $Z_n$. This is obvious from excluding every $i$-th component, like first here:
$b_1 a_2 x_2 + b_1 a_3 x_3 + ... = 0 \\ a_1 b_2 x_2 + a_1 b_3 x_3 + ... = 0$
So I was thinking, what is such as elegant expression for n-dimensional intercrossing of two planes?