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 .
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.