Angle between two planes in $n$-dimensional space, where $n \geq 4$

Question

As known, an angle between two planes in $3$d is an angle between normals to them. But how about $n$-dimensional spaces, $n\geq4$. Is it the same for them? Why is it so?

I have two planes, given parametric in a $4$D Euclidean space ( $U: a_{0}+a_{1}t_{1}+a_{2}t_{2}$ and V: $b_{0}+b_{1}t_{1}+b_{2}t_{2}$, where $a_{i}$, $b_{i}$ are given $4$D vectors) and I need to find an angle between these planes.

I think that the task need to be done the following way:

1. ortogonalise $a_{i}$ in U and $b_{i}$ in V
2. take a random vector that is not in U or V and find normals to every plane
3. find an angle between these two normals using a formula: $\dfrac{(n_{u}, n_{v})}{||n_{u}||||n_{v}||}$,

Is it right?. In my class textbooks or via Internet I could not find anything relevant to such a case.