To ensure that the game has a single, completely-mixed, Nash equilibrium there needs to be "cyclic unique best responses", i.e., either
BR(L) = O, BR(O) = R, BR(R) = U, BR(U) = L,
BR(L) = U, BR(U) = R, BR(R) = O, BR(O) = L.
The strict cyclic preferences both rule out pure equilibria and allow a (unique) completely mixed equilibrium.
There are infinitely many ways to achieve either of the two cases above. Aiming for the first case with BR(L) = O, given that the payoff to the row player for (O,L) is 1, to ensure that BR(L) = O, i.e., the unique best response against L is O, one can set the payoff to the row player for (U,L) as any number strictly less than 1, i.e. so that against L, the row player strictly prefers O with payoff 1 to U with payoff strictly less than 1. The rest of the payoffs can be filled in similarly.