Skew symmetric matrices even size commutativity

Given Data in the question

1. $w(t)=\frac{1}{2}\begin{bmatrix} 0 &r(t) &-q(t) &p(t) \\ -r(t)& 0 &p(t) &q(t) \\ q(t)& -p(t) &0 & r(t)\\ -p(t)&-q(t) &-r(t) &0 \end{bmatrix}\tag1$

2. $B(t)= -\int_{0}^{t} w(t) \ dt \tag 2$

Question

1. Can we clearly prove $B(t)w(t)=w(t)B(t)$. Means can we prove the commutative property between these two? 2. What is $\frac{\mathrm{d} }{\mathrm{d} t}{e^{B(t)}}$?

NB :: Remember this question is different from skew symmetric matrices of size 3. Because skew symmetric matrices of odd size has zero determinant. Here it is even size