matrices – How to design a matrix function to meet the following conditions

Suppose I have a matrix $$Xin mathbb{R}^{ntimes n}$$, such that

1. $$X$$ is symmetric
2. Do not know the rank

I want design a matrix function $$f(X,Q)in mathbb{R}^{ntimes n}$$ with $$Q = qq^T$$ and $$qin mathbb{R}^n$$, such that

1. $$f(X,Q)$$ is symmetric
2. $$text{tr} (f(X,Q))=0$$
3. rank$$(f(X,Q))$$ is $$2$$.

For

1. we can design $$QX+XQ$$
2. I only know the property $$Omega Q + QOmega^T$$ with $$Omega$$ a skew-symmetric matrix (since tr$$(QOmega)=q^TOmega q=0$$.
3. Choose $$f(X,Q)=QX+XQ$$. The basis for its column space is $${q, Xq}$$

However, I have no idea how to combine all of them. Can anyone help me this?

Sincerely appreciate this help.