Definition
Let $V$ be a vector space and $k$ a non-negative integer. Given $T\in\mathcal L^k(V)$ and $\vec v\in V$ let $i_{\vec v}T$ be the $(k-1)$-tensor which takes the value $$ \big[\iota_{\vec v}(T)\big](\vec v_1,...,\vec v_{k-1}):=\sum_{r=1}^k(-1)^{r-1}T(\vec v_1,...,\vec v_{r-1},\vec v,\vec v_r,...,\vec v_{k-1}) $$ for any $\vec v_1,..,\vec v_{k-1}\in V$.
So it is not hard to show that the operation above defined is linear with respect $\vec v$ and $T$ and moreover it is not hard to show that $$ \iota_{\vec v}\big(\vec v^1\otimes...\otimes\vec v^k\big)=\sum_{r=1}^k(-1)^{r-1}v^r(\vec v)\cdot\vec v^1\otimes...\otimes\vec v^{r-1}\otimes\vec v^{r+1}\otimes...\otimes\vec v^k $$ for any $\vec v^1,...,\vec v^k\in V^*$ and thus using this I tried to prove that $$ \iota_{\vec v}(\vec v^1\wedge...\wedge\vec v^k)=\sum_{r=1}^k(-1)^{r-1}v^r(\vec v)\cdot\vec v^1\wedge...\wedge\vec v^{r-1}\wedge\vec v^{r+1}\wedge...\wedge\vec v^k $$ but unfortenately I did not be able to do it. So could someone helpme, please?
To follow a my proof attempt.
MY PROOF ATTEMPT
To follow I use Laplace formula with respect the $r$-th column
$$ \big[\iota_{\vec v}(\vec v^1\wedge...\wedge\vec v^k)\big](\vec w_1,...,\vec w_{k-1}):=\sum_{r=1}^k(-1)^{r-1}\big[\vec v^1\wedge...\wedge\vec v^k\big](\vec w_1,...,\vec w_{r-1},\vec v,\vec w_r,...,\vec w_{k-1})=\\ \sum_{r=1}^k(-1)^{r-1}\det\begin{pmatrix}v^1(\vec w_1)&&\cdots&&v^1(\vec w_{r-1})&&v^1(\vec v)&&v^1(\vec w_r)&&\cdots&&v^1(\vec w_{k-1})\\ \vdots&&\ddots&&\vdots&&\vdots&&\vdots&&\ddots&&\vdots\\ v^k(\vec w_1)&&\cdots&&v^k(\vec w_{r-1})&&v^k(\vec v)&&v^k(\vec w_r)&&\cdots&&v^k(\vec w_{k-1})\end{pmatrix}=\\ \sum_{r=1}^k(-1)^{r-1}\sum_{h=1}^k(-1)^{h+r}v^h(\vec v)\det\begin{pmatrix}v^1(\vec w_1)&&\cdots&&v^1(\vec w_{r-1})&&v^1(\vec w_r)&&\cdots&&v^1(\vec w_{k-1})\\ \vdots&&\ddots&&\vdots&&\vdots&&\ddots&&\vdots\\ v^{h-1}(\vec w_1)&&\cdots&&v^{h-1}(\vec w_{r-1})&&v^{h-1}(\vec w_r)&&\cdots&&v^{h-1}(\vec w_{k-1})\\ v^{h+1}(\vec w_1)&&\cdots&&v^{h+1}(\vec w_{r-1})&&v^{h+1}(\vec w_r)&&\cdots&&v^{h+1}(\vec w_{k-1})\\ \vdots&&\ddots&&\vdots&&\vdots&&\ddots&&\vdots\\ v^k(\vec w_1)&&\cdots&&v^k(\vec w_{r-1})&&v^k(\vec w_r)&&\cdots&&v^k(\vec w_{k-1})\end{pmatrix}=\\ \sum_{r=1}^k\sum_{h=1}^k(-1)^{r-1}(-1)^{h+r}v^h(\vec v)\big[\vec v^1\wedge...\wedge\vec v^{h-1}\wedge\vec v^{h+1}\wedge...\wedge\vec v^k\big](\vec w_1,...,\vec w_{k-1})=\\ \sum_{r=1}^k\sum_{h=1}^k(-1)^{h-1}v^h(\vec v)\big[\vec v^1\wedge...\wedge\vec v^{h-1}\wedge\vec v^{h+1}\wedge...\wedge\vec v^k\big](\vec w_1,...,\vec w_{k-1})=\\ k\cdot\sum_{h=1}^k(-1)^{h-1}v^h(\vec v)\big[\vec v^1\wedge...\wedge\vec v^{h-1}\wedge\vec v^{h+1}\wedge...\wedge\vec v^k\big](\vec w_1,...,\vec w_{k-1}) $$
So as you can see the formula is different only of the factor $k$ but unfortunately I do not see where is my mistake: so could you help me, please?