How can one compute each of the following matrices, explicitly:

$$\int_{O(n)} e^{g}dg$$ or $$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$ What is the explicite entries of the resulting matrices, for $n=2$?

Moreover, for $n,m\in \mathbb{Z}$ define the linear operator $T_{n,m}$ on $M_{n}(\mathbb{R})$ as follow: $$T_{n,m}(A)=\int_{O(n)}g^{n}Ag^{m}$$ Under what suficcient and necessary condition, $T_{n,m}$ is conjugate to $T_{n',m'}$? Is there an explicit formulation for $T_{n,m}$?

The integration is based on the Haar measear defined on orthogonal group $O(n)$.