L= { \
{  0,  0,    a,  0,    0,  0,    0,  0,    0,  0,    0,  0   }, \
{  0,  0,    0, -a,    0,  0,    0,  0,    0,  0,    0,  0   }, \
\
{ -a,  0,    0,  b,    a,  0,    0,  0,    0,  0,    0,  0   }, \
{  0,  a,   -b,  0,    0, -a,    0,  0,    0,  0,    0,  0   }, \
\
{  0,  0,   -a,  0,    0,  b,    a,  0,    0,  0,    0,  0   }, \
{  0,  0,    0,  a,   -b,  0,    0, -a,    0,  0,    0,  0   }, \
\
{  0,  0,    0,  0,   -a,  0,    0,  b,    a,  0,    0,  0   }, \
{  0,  0,    0,  0,    0,  a,   -b,  0,    0, -a,    0,  0   }, \
\
{  0,  0,    0,  0,    0,  0,   -a,  0,    0,  b,    a,  0   }, \
{  0,  0,    0,  0,    0,  0,    0,  a,   -b,  0,    0, -a   }, \
\
{  0,  0,    0,  0,    0,  0,    0,  0,   -a,  0,    0,  0   }, \
{  0,  0,    0,  0,    0,  0,    0,  0,    0,  a,    0,  0   } \
\
};
Simplify[Det[L]]
TeXForm[%]