Let $\displaystyle (X_i)_{i\geq 0}$ be a family of independent $\displaystyle n$-dimensional standard Gaussian random variables (or any other continuous distribution supported by $\displaystyle \mathbb{R}^n$). Almost surely, this family answers your question.
Density: comes as a direct consequence of Borel-Cantelli lemma (for any ball $\displaystyle B$, $\displaystyle \forall i,\ P(X_i\in B)=p>0$ hence $\displaystyle \sum_i P(X_i\in B)=\infty$, and these events are independent, so that infinitely many terms of the sequence fall in $\displaystyle B$ almost-surely)
Non-colinearity: for any distinct $\displaystyle i,j,k$, $\displaystyle P(X_k\in (X_i X_j))=0$ because the line $\displaystyle (X_i X_j)$ has zero Lebesgue measure (since $\displaystyle n\geq 2$), and $\displaystyle X_k$ has a density, and is independent of $\displaystyle X_i,X_j$. There are countably many triplets $\displaystyle i,j,k$, hence the probability that there exists one satisfying the alignment property is 0. (A union of countably many sets of zero measure has zero measure).