Analog of $\operatorname{Var}[X] = E[X^2] - E^2[X] $ for random matrices

Question

For a scalar random variable, we have the identity: $\operatorname{Var}[X] = E[X^2] - E^2[X] $

Is there a similar formula for $X$ being a random matrix? For sure, the covariance "matrix" would be a 4-dimensional tensor.