Let $X \in \mathfrak {X}^{k} (M).$ Then $X \in \Gamma (\wedge^{k} TM)$ i.e. $X: M \longrightarrow \wedge^{k} T M$ is a smooth map such that $X_{p} \in \wedge^{k} T_{p} M$ (i.e. $X_{p}$ is a alternating $k$-tensor) for each $p \in M.$ In the lecture notes I am following it has been argued (without being explicit) we can view it as $C^{\infty} (M)$-multilinear, alternating linear map from $\Omega^{1} (M) \times \cdots \times \Omega^{1} (M)\ (k \text {-times}) \longrightarrow C^{\infty} (M).$
But how is it defined? My intuition suggests that the map takes the following form $:$
$$(\omega_{1}, \cdots, \omega_{k}) \mapsto (p \mapsto (\omega_{1,p} \otimes \cdots \otimes \omega_{k,p}) (X_{p})).$$ But I find difficulty in showing equivalence of two definitions of a multivector field if I use this definition. What I mean to say is $:$ Suppose we have a $C^{\infty} (M)$-multilinear, alternating linear map $X : \Omega^{1} (M) \times \cdots \times \Omega^{1} (M)\ (k \text {-times}) \longrightarrow C^{\infty} (M).$ Can I associate a $k$-multivector field corresponding to that if we use the above definition?
Your intuition is basically all you need.
Given your $C^\infty(M)$-multilinear alternating map $X:\Omega^1(M)\times\ldots\times\Omega^1(M)\to C^\infty(M)$, the induced multivector field $\tilde X:M\to \wedge^k TM$ sends $p\in M$ to the element $\tilde X_p\in\wedge^k T_pM$ that is uniquely characterized by the property $$ \langle \omega_p^1\wedge\ldots\wedge\omega_p^k,\tilde X_p \rangle = \left(X(\omega_p^1,\ldots,\omega_p^k)\right)(p), \quad \text{for all } \omega_p^i\in T_p^*M, $$ where $\langle \cdot\,,\cdot \rangle$ is the natural pairing of a vector space with its dual.
Your question boils down to the following linear algebra fact: Given a finite dimensional vector space $V$, then for $v,w\in V$ we have $$ \langle \alpha,v \rangle = \langle \alpha,w \rangle,\quad \text{for all } \alpha \in V^* \qquad \Longleftrightarrow \qquad v=w $$