We have the following beautiful result for Pauli $su(2)$ matrices

$$(\vec{\sigma}\cdot\vec{a})(\vec{\sigma}\cdot\vec{b}) = \mathbb{I} ~\vec{a}\cdot\vec{b} + i (\vec{a} \times \vec{b}) \cdot \vec{\sigma}.$$

Do we have a similar structure for [Gell-Mann $su(3)$ matrices](https://en.wikipedia.org/wiki/Gell-Mann_matrices)? Specifically, what would the following be

$$(\vec{\lambda}\cdot\vec{a})(\vec{\lambda}\cdot\vec{b}) = ~?$$