# 2-vector fields on smooth manifolds

I have some questions concerning smooth 2-vector fields on smooth manifold $M$. By definition they are smooth sections of $TM \otimes TM$.

1. At first, in local coordinates every 2-vector field is a linear combination with smooth coefficients of basic 2-vector fields $\partial_i \otimes \partial_j$. Since we can define for any smooth functions $f$, $g$ on M the value $\partial_i \otimes \partial_j (f,g) = \partial_i f \partial_j g$ it seems that $\Gamma(TM \otimes TM)$ is a space of smooth bilinear mappings from $C^\infty(M)$ to $C^\infty(M)$ satisfying the Leibniz rule with respect to each variable. Is there a name for such objects?
2. Is there an easy way to deal with these objects to obtain coordinate-free expressions? For example, if we want to obtain coordinate-free expression for the Lie derivative of vector field $Y$ in direction of vector field $X$ we consider $Y$ as a derivation on $С^\infty(M)$; if we want to obtain the coordinate-free expression for the Lie derivative of a $k$-covector field we consider it as a $k$-linear mapping from $\Gamma(TM)$ to $C^\infty(M)$. But if I want to obtain such expression for a $2$-vector, I don't know if I should consider it as a bilinear mapping from $C^\infty(M)$ to $C^\infty(M)$ or as a bilinear mapping from $\Gamma(T^*M)$ to $C^\infty(M)$. In both cases I don't know how to compute $(\phi^X_{-t})_* \alpha_{\phi^X_t(p)}$, where $\alpha$ is a $2$-vector, $\phi$ is a flow generated by $X$, $p \in M$.
3. If $X$ is a smooth vector field then for any smooth function $f$ we have $$f\bigl(\phi^X_t(p)\bigl) = f(p)+X_p(f)t+o(t).$$ This equality enables me to obtain a lot of goods considering vector fields as derivations on $C^\infty(M)$. Is there some analogic formula for $2$-vector fields?
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Just a nitpick: In 2. I think you're asking how to compute it for a 2-vector, not for a 2-covector. – gofvonx Nov 22 '13 at 18:04
@gofvonx yes, thanks. I would note that that 2 is only about coordinate-free expressions. – Nimza Nov 22 '13 at 19:44