We define equivariant tensors for every non-negative integer $p$ and every Weil algebra $A$ and establish a one-to-one correspondence between the equivariant tensors and linear natural operators lifting skew-symmetric tensor fields of type $(p,0)$ on an $n$-dimensional manifold $M$ to tensor fields of type $(p,0)$ on $T^AM$ if $1\le p\le n$. Moreover, we determine explicitly the equivariant tensors for the Weil algebras ${\mathbb D}^r_k$, where $k$ and $r$ are non-negative integers.
We establish a formula for the Schouten-Nijenhuis bracket of linear liftings of skew-symmetric tensor fields to any Weil bundle. As a result we obtain a construction of some liftings of Poisson structures to Weil bundles.