I'm reading this paper about the c-map between special Kähler manifolds and Hyperkähler manifolds and in the introduction the authors talk about the cotangent bundle as a certain associated bundle of a principal GL(M)-bundle. Hence, the consider a connection $1$-form $\omega$ which, by definition, satisfies some properties.
The problem is that the authors use a technical language I'm not used to. So I've tried to write $\omega$ acting over vector fields. My procedure is as follows:
When I studied linear connections, I undesrtood the connection $1$-form as $n^2$ $1$-forms instead of an $\operatorname{End}(TM)$-valued $1$-form. In components, those $1$-forms read as
$$ \omega_i^j = \Gamma_{ik}^j \theta^k . $$
So, what I want to do is to define $\omega$ as $\operatorname{End}(TM)$-valued $1$-form using the above $\omega_i^j$.
Hence, for each vector field $X$, I define $\omega(X):TM\rightarrow TM$ as
$$ \omega(X)(Y)=\nabla_XY - X(Y^j)e_j .$$
This expression is $C^\infty(M)$-linear in $X$ trivially but also in $Y$, so it might be valid. And that is what I ask to you. IS the above expression a correct definition for a connection $1$-form? And, It would be possible to get a similar expression but without coordinates (I would like to remove the term $X(Y^j)e_j$?
Thanks
EDIT: Let me clarify my notation. $\nabla$ refers to a linear connection. I think it should be a connection on the principal bundle but since the paper is focus on special Kähler manifolds I think it will reduce to a linear connection at the end. $\Gamma_{in}^k$ denotes the symbols of the connection, the Christoffel symbols. Finally, ${e_i}$ is a frame and ${\theta^j}$ is the dual frame.