Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and $\pi:P\to M$ a principal $G$-bundle. A connection $A$ on $P$ is a $\mathfrak{g}$-valued one-form, i.e. an element of $\Omega^1(P)\otimes\mathfrak{g}$, satisfying some properties. The curvature $F$ of $A$ is an element of $\Omega^2(P)\otimes\mathfrak{g}$ defined by $$F(X,Y)=dA(X^H,Y^H)$$ where $X,Y$ are vector fields on $P$ and $X^H,Y^H$ their horizontal components.
Some texts say that the curvature can be seen as an element of $\Omega^2(\mathrm{ad}(P))$ where $\mathrm{ad}(P):=P\times_G\mathfrak{g}$ is the vector bundle associated to the adjoint representation. How to see this? What element of $\Omega^2(P\times_G\mathfrak{g})$ precisely? And how can we recover the original curvature from it?
The connection $A$ is a tensorial $1$-form i.e $g\in G, R_g^*A=Ad(g^{-1})A$. This implies that the covariant derivative of $A$ which is the curvature is a tensorial 2-form. i.e $R_g^*F=Ad(g^{-1})F$. This is equivalent to saying that $F\in \Omega^2(P\times_G\mathfrak{g})$. To see this, consider the trivialization $(U_i\times G)$ of $P$. Suppose that the transition functions of $P$ are defined by $u_{ij}$, the transition functions of $P\times_G\mathfrak{g}$ are defined by $Ad(u_{ij}^{-1})$, Let $x\in U_i\cap U_j\times G, u,v\in T_x(U_i\cap U_j\times G)$, the transition function sends $x$ to $xu_{ij}$ and $u$ to $R_{u_{ij}}^*u$ we deduce that $F_{xu_{ij}}(R_{u_{ij}}^*u,R_{u_{ij}}^*v)=Ad(u_{ij}^{-1})F(u,v)$ since $F$ is tensorial. This is the local definition of an element of $\Omega^2(P\times_G\mathfrak{g})$.