U(n) v.s. SO(n) and their quotient / homogeneous spaces

Question

By counting the number of generators, it is easy to see that unitary group U(n) has much more generator than SO(n).

1. So it would make sense to consider the mod out group or the quotient space of U(n)/SO(n).

   • What is the quotient space U(10)/SO(10)?

   • What is the quotient space U(16)/SO(16)?

2. On the other hand, we can consider possibly $$ 1 \to N \to \text{U}(n) \to \text{SO}(n)\to 1, $$ such that $N$ is some proper normal subgroup of a topological group, such that

   • $ \text{U}(10) /N=\text{SO}(10)=? $

   • $ \text{U}(16) /N=\text{SO}(16)=? $

What are the $N$ group/space?

You may get my questions but find my descriptions imprecise, so please help on answering and clarifying the question to an appropriate manner.