Space of Alternating $k$-Tensors Notation

In Lee's 'Intro to Smooth Manifolds', $\Lambda^k(V)$ refers to the space of alternating $k$-tensors on a vector space $V$, as you mentioned. However, the space $\Omega^k(M)$ is the space of smooth $k$-forms on a smooth manifold $M$. That is, an element of $\omega \in \Omega^k(M)$ is a smooth map $M \to \Lambda^k(T^* M)$ (called a smooth section of of the bundle $\Lambda^k(T^* M)$), so for each point $x \in M$, we get an alternating $k$-tensor $\omega(x) \in \Lambda^k(T^* M)$. This space is often written as $\Omega^k(M) = \Gamma(\Lambda^k(T^* M))$.

Not entirely sure however what $\Omega^k(V)$ is, when $V$ is just a vector space.