# Does every element in exterior algebra have the form $v_1 \wedge v_2 \wedge … \wedge v_k$? [duplicate]

Does every element in $\wedge ^k V$ can be expressed as the form $v_1 \wedge v_2 \wedge ... \wedge v_k$ ? Here $V$ is a n-dim vector space, and $v_i$ are vectors in $V$.

Intuitively it is right, but I have problem in how to make the sum of two forms of $v_1 \wedge v_2 \wedge ... \wedge v_k$ still have this form .

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This question was marked as an exact duplicate of an existing question.

No. As a counterexample, take $V=\Bbb R^4$ with basis $e_1,\dots,e_4$. Note that the vector $$e_1 \wedge e_2 + e_3\wedge e_4 \in \wedge^2V$$ cannot be written in such a fashion.