Symplectic orbits of unimodular rows

For a smooth affine algebra $R$ of dimension $d\ge 3$ over a field $k$ and an invertible alternating matrix $\chi$ of rank $2n$, the group $Sp(\chi )$ of invertible matrices of rank $2n$ over $R$ which are symplectic with respect to $\chi$ acts on the right on the set $U{m}_{2n}(R)$ of unimodular rows of length $2n$ over $R$. In this paper, we prove that $Sp(\chi )$ acts transitively on $U{m}_{2n}(R)$ if $k$ is algebraically closed, $d!\in k^{\times }$ and $2n \geq d$.