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 $n=2$?

The integration is based on  Haar measear