The subspace $L((x_1\wedge\dots\wedge x_k)^m)$ of $S^m(\wedge^k\mathbb R^n)$

Let $\wedge^k\mathbb R^n$ be the $k$th outer power of a space $\mathbb R^n$, $V(m,n,k)=S^m(\wedge^k\mathbb R^n)$ the $m$th symmetric power of $\mathbb R^n$, and $V_0=L((x_1\wedge\dots\wedge x_k)^m):x_i\in\mathbb R^n$). We construct a basis and compute a dimension of $V_0$ for $m=2$, and for $m$ arbitrary, present an effective algorithm of finding a basis and computing a dimension for the space $V_0(m,n,k)$. An upper bound for the dimension of $V_0$ is established, which implies that
$$ \lim_{m\to1}\frac{\dim V_0(m,n,k)}{\dim V(m,n,k)}=0. $$
The results obtained are applied to study a Grassmann variety and finite-dimensional Lie algebras.