For an $n$-dimensional Lie algebra, is there always a matrix representation $\{M\}$ and a single vector $v$ such that $\{Mv\}$ is $n$-dimensional?

This would necessarily be a faithful representation.

I'm focusing on Lie algebras over $\mathbb R$, but more general answers are welcome.

This might have something to do with weights or Whitehead's lemma, but I don't know enough about representation theory to be sure.

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