Let $V$ be a real $n$-dimensional vector space, and let $W \le \bigwedge^k V$ be a subspace . Suppose that $\dim W \ge 2$. Does $W$ contain a non-zero decomposable element?
If $\dim W=1$, then clearly we can take $W=\text{span} (\sigma)$ for some non-decomposable $\sigma \in \bigwedge^k V$.
No. Consider the $3$-dimensional subspace $W\subset\Lambda^2\mathbb R^4$ spanned by
$$B_1=e_2e_3+e_1e_4,\;B_2=e_3e_1+e_2e_4,\;B_3=e_1e_2+e_3e_4$$
where $e_i$ span $\mathbb R^4$. See that $B_iB_j=0$ for $i\neq j$, and $B_iB_i=2e_1e_2e_3e_4$.
Thus, for any $X\in W\setminus\{0\}$,
$$X=x_1B_1+x_2B_2+x_3B_3$$
$$XX=2e_1e_2e_3e_4(x_1^2+x_2^2+x_3^2)\neq0$$
which shows that $X$ is not decomposable.
These bivectors correspond to isoclinic rotations in $SO(\mathbb R^4)$.