In Hamilton's Mathematical Gauge Theory he gives the following definition.
Let $V$ be a finite-dimensional $\mathbb{K}$-vector space with a symmetric bilinear form $Q$. For $v \in V$ and $\sigma \in \Lambda^k V$ there is a unique $(k-1)$-form $v \lrcorner \sigma \in \Lambda^{k-1}V$, called the contraction of $v$ and $\sigma$, with the following properties
if $\sigma \in V$, then $v \lrcorner = Q(v, \sigma)$;
For all $\sigma \in \Lambda^kV, \omega \in \Lambda^\ell V$ we have $$v \lrcorner(\sigma \wedge \omega) = (v \lrcorner \sigma) \wedge \omega + (-1)^k \sigma \wedge (v \lrcorner \omega).$$
I am familiar with contractions if we take $V = V^*$, in which case $\Lambda^kV^*$ is simply the set of differential forms of degree $k$. Thus it makes sense to contract this object with a vector (i.e. interior multiplication).
However the above definition is more general. An element of $\Lambda^k$ is an alternating tensor of vectors and not covectors. That is, an element of $\Lambda^k$ is of the form $v_1 \wedge \cdots \wedge v_k$. These cannot take a vector as an input, so how does it make sense to contract them?