Matrix Algebra - Linear dependency

Question

We have a given equation

$ \frac{\mathrm{d}R(t) }{\mathrm{d} t}=R(t) \{(1-t)U_0+t U_1\}\tag 1$,

all variables except scalar variable 't' has dimension $3 \times 3$.

Given data

1. $R(t)$ is a rotation matirx(Determinant 1, orthogonal etc,orthonormal,etc). We know what is $R(0)$
2. Given $U_0,U_1$ are skew symmetric matrices and are constant
3. $t\in[0,1]$
4. Any rotation matrix can be written as a matrix exponent of skew symmetric matrix. So $R(t)=e^{B(t)}$,where B(t) is a skew symmetric matrix. We already know the relationship between $R(t)$ and interpolated value of $U_0,U_1$ in equation 1. Assume we have given the value of $B(0)$ and $B(1)$

Question

1. Can we find a relationship between B(t) and $U_0,U_1$? means can we write B(t) as a function of t and $U_0,U_1$ only
2. Is it possible to write $R(t)=e^{B(0)(1-t)+tB(1)}$, if so how can we prove it?