Is there a generalized equation for a line in barycentric coordinates?

Question

I've read that in the plane the equation for points $\lambda=(\lambda_1,\lambda_2,\lambda_3)$ on the line passing through $\mu=(\mu_1,\mu_2,\mu_3)$ and $\nu=(\nu_1,\nu_2,\nu_3)$ is $$ \left|\begin{array} \\ \mu_1 & \mu_2 & \mu_3 \\ \nu_1 & \nu_2 & \nu_3 \\ \lambda_1 & \lambda_2 & \lambda_3 \end{array}\right|=0 $$ And I want a formula that generalizes this to higher dimensions. It seems most natural to assume that the generalization is $$\mu\wedge\nu\wedge\lambda=0$$ If this is the case I'd like hints on how to prove it, otherwise I'd just like to know whether or not a neat generalization exists and what it is.