Let $V$ be $k^{n}$, where $k$ is an algebraically closed field. Then we can compute the homogeneous coordinates of $\mathbb{P}V$ as follows: for a point $p\in \mathbb{P}V$ ($p$ is a line in $V$), take the coordinate (coordinate in $V$) of a non-zero point of $p$.
If $V$ is replaced by $\Lambda^{s}V$, how to compute the homogeneous coordinates of points in $\mathbb{P}(\Lambda^{s}V)$? Thank you very much.
Edit: Let $G(s, V)$ be the set of subspaces of $V$ of dimension $s$. There is a map: $\phi: G(s, V) \to \mathbb{P}(\Lambda^{s}V)$ defined by sending $M\in G(s, V)$ to the homogeneous coordinate of $M$ in $\mathbb{P}(\Lambda^{s}V)$. By identify $V$ with $k^{n}$, $M$ can be represented by a $k\times n$ matrix. Why the homogeneous coordinate of $M$ are minors of the matrix.
Edit: I know the answer now.