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Given v={v1,v2,v3,v4} how can i construct the matrix

 M={{ v1, v2, v3, v4, -v3, -v2},
    { v2, v1, v2, v3,  v4, -v3},
    { v3, v2, v1, v2,  v3,  v4},
    { v4, v3, v2, v1,  v2,  v3},
    {-v3, v4, v3, v2,  v1,  v2},
    {-v2,-v3, v4, v3,  v2,  v1}}
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Looks like ToeplitzMatrix can be useful for this case:

 ToeplitzMatrix[{v1, v2, v3, v4, -v3, -v2}]
{{ v1,  v2, v3, v4, -v3, -v2}, 
 { v2,  v1, v2, v3,  v4, -v3}, 
 { v3,  v2, v1, v2,  v3,  v4}, 
 { v4,  v3, v2, v1,  v2,  v3}, 
 {-v3,  v4, v3, v2,  v1,  v2}, 
 {-v2, -v3, v4, v3,  v2,  v1}}

There is also an undocumented option "ReflectedNegation" for ArrayPad which can help to generate the reflected input array i.e.:

v = {v1, v2, v3, v4};
ArrayPad[v, {0, 2}, "ReflectedNegation"]
{v1, v2, v3, v4, -v3, -v2}

Thus, the full command will be:

ToeplitzMatrix[ArrayPad[v, {0, 2}, "ReflectedNegation"]]
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