We are given vectors $\mathbf a_1,\dots,\mathbf a_N\in\mathbb C^n$. Let these vectors be columns of $n$-by-N matrix $\mathbf A$. We additionally know that any $n-1$ vectors are linearly independent. This makes possible to do the following: we consider all $n-1$ tuples $\mathbf a_{i_1},\dots,\mathbf a_{i_{n-1}}$ (there are $N$ choose $n-1$ of them) and for very $n-1$-tuple we take the vector (it is unique up to scaling factor) which is orthogonal to $\mathbf a_{i_1},\dots,\mathbf a_{i_{n-1}}$.

Let the new vectors form the matrix $\mathbf B$. Then $\mathbf B$ is $n$-by-$N$ choose $n-1$ matrix. (We also assume that the columns of $\mathbf B$ are ordered somehow)

The question is: is there exist some special name and the notation for the matrix $\mathbf B$? (some one told me that this operation relates somehow to the Hodge star dual)