So I am trying to derive a nice general formula for $Cov(X,Y,Z)$ and $Corr(X,Y,Z)$, I defined it as such $$ Cov(X,Y,Z) = E[(X-E[X])(Y-E[Y])(Z-E[Z])] $$ $$ Corr(X,Y,Z) = \frac {Cov(X,Y,Z)} {\sqrt{Var[X]Var[Y]Var[Z]}} $$
Now doing some algebra got me to this expression: $$ Cov(X,Y,Z) = E[XYZ]-E[Z]E[XY]-E[X]E[ZY]-E[Y]E[XZ]+2E[X]E[Y]E[Z] $$
When I apply this formula however, my correlation is sometimes over 1 and below -1.
I have two questions:
1) Is the covariance ever defined this way (because I notice people always refer to the covariance matrix)?
2) Was my algebra wrong, or is the property $ |COV|\le1 $ is not transferable to 3 r.v.s?