Does this ODE have an explicit solution? How can it be characterized?

Question

Consider this ODE for $x\in\mathbb{R}^2$ $$ \frac{ d x}{d t} = \pmatrix{\cos(t) & -\sin(t)\\ \sin(t) &\cos(t)} \pmatrix{0 & 0\\ 0 & -1} \pmatrix{\cos(t) & \sin(t)\\ -\sin(t) &\cos(t)} x $$ Does it have an explicit solution? Which qualitative properties does it have?