Decomposition of a matrix into a skew symmetric matrix and "positive definite" symmetric matrix

Question

I have a special case which is slightly different from the usual decomposition of a matrix $A$ into skew symmetric and symmetric matrices ($A=\frac{1}{2}(A-A^T)$+ $\frac{1}{2}(A+A^T)$).
Here I need to have the symmetric part to be "POSITIVE Definite" as well.

Is there another decomposition technique which guarantees to have the symmetric matrix also positive definite?