What is the exterior derivative of a connection form?

Question

What is the exterior derivative of a connection form?

Given a principal $G$-bundle $P$ with base space $B$, we assume that $G$ is a Lie group and $P,B$ are smooth manifold. A connection form is a Lie algebra valued 1 form \begin{equation} A=\sum_{i=1}^kA_i\otimes X_i\in\Omega^1(P)\otimes \mathfrak g \end{equation} such that

1. $A$ is $G$-invariant with repect to the product action of $G$ on $P$ and $\mathfrak g$
2. $A$ is vertical, i.e. $A(X^\#)=X$ for $X\in\mathfrak g$, where $X^\#$ is the vector field over $P$ induced by $X\in\mathfrak g$

My question is: what is $dA$? It seems that $dA\in\Omega^2(P)\otimes\mathfrak g$ and \begin{equation} dA(X,Y)\xrightarrow{?}X\cdot A(Y)-Y\cdot A(X)-A([X,Y]) \end{equation} then what is $X\cdot A(Y)$?

Thanks in advance.