I wish to draw uniformly a vector that has a given fixed angle to a predetermined vector. I am looking for a general formula for any space of dimensions $\geq2$ (because angles in a one-dimensional space does not make much sense).
In other words, given a fixed angle $\theta$ and an initial vector $u$, I wish to sample uniformly from the set
$$\{v\in\mathbb{R}^N| \arccos{\frac{u^Tv}{\|u\|\|v\|}}=\theta, \|v\|=1\}$$
Let $e_N$ be the $N^{\rm th}$ element of the standard basis of $\mathbb R^N$, and identify $\mathbb R^{N-1}$ with the subspace of $\mathbb R^N$ orthogonal to $e_N$. Let $S$ be the unit sphere in $\mathbb R^{N-1}$.
Let $T$ be any orthogonal transformation sending $e_N$ to $u/\|u\|$ (concretely you can find such a transformation using Gram-Schmidt). Then your set is exactly $$ T((\cos\theta)e_N+(\sin\theta)S). $$ So it suffices to sample uniformly from $S$. One way is to sample from an $(N-1)$-dimensional Gaussian and normalize (see this question).