In the literature I can only find ChernSimons terms for odddimensional manifolds. For example, for a $G$bundle over a 3dimensional manifold we have $A \wedge dA + A \wedge A \wedge A$ with $A$ being a $\mathfrak{g}$valued 1form. Why can't I write such forms for evendimensional manifolds?

It has to do with the fact that the characteristic classes (over the reals) of a principal $G$bundle have even degree. We can associate ChernSimonslike theory to each characteristic class of degree $2k$ together with a $G$bundle $P$ over a manifold of dimension $2k1$. To be a bit more technical a ChernSimonslike form is asssociated to the following data 1. A homogeneous polynomial $\Phi$ of degree $k$ on the Lie algebra of $G$ invariant under the action of $G$ by conjugation. 2. A principal $G$bundle $P\to M$ over $M$. 3. A pair of connections $\nabla^0, \nabla^1$ on $P\to M$. The ChernWeil theory produces two closed forms $$ \Phi(\nabla^0),\Phi(\nabla^1)\in \Omega^{2k}(M) $$ and a form $$ T\Phi(\nabla^1,\nabla^0)\in \Omega^{2k1}(M), $$ such that $$ d T\Phi(\nabla^1,\nabla^0)= \Phi(\nabla^1)\Phi(\nabla^0). $$ (For details see Chapter 8 of these notes.) The transgression form $T\Phi(\nabla^1,\nabla^0)$ is the one used in ChernSimons theories. It depends on two connections, but usually $\nabla^0$ is some fixed connection. 

