How to construct the following matrix

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Here are a few ways to produce this matrix. One could think of many more ways.

I'll use $n = 5$ to save space. Adapt to $n = 10$.

sage: n = 5

This may be the more efficient:

sage: matrix.circulant([0, 1] + [0] * (n - 3) + [-1])
[ 0  1  0  0 -1]
[-1  0  1  0  0]
[ 0 -1  0  1  0]
[ 0  0 -1  0  1]
[ 1  0  0 -1  0]

Using a lambda function:

sage: a = matrix(ZZ, n, lambda i, j: ((j - i) % n == 1) - ((i - j) % n == 1))
sage: a
[ 0  1  0  0 -1]
[-1  0  1  0  0]
[ 0 -1  0  1  0]
[ 0  0 -1  0  1]
[ 1  0  0 -1  0]

More by hand:

sage: a = matrix(ZZ, n, [[((j - i) % n == 1) - ((i - j) % n == 1) for j in range(n)] for i in range(n)])
sage: a
[ 0  1  0  0 -1]
[-1  0  1  0  0]
[ 0 -1  0  1  0]
[ 0  0 -1  0  1]
[ 1  0  0 -1  0]

A variation, maybe slower because of the matrix subtraction.

sage: b = matrix(ZZ, n, [[(j-i)%n == 1 for j in range(n)] for i in range(n)])
sage: a = b - b.T
sage: a
[ 0  1  0  0 -1]
[-1  0  1  0  0]
[ 0 -1  0  1  0]
[ 0  0 -1  0  1]
[ 1  0  0 -1  0]

One might also prefer to produce a sparse matrix.

sage: b = matrix(ZZ, n, {(i, (i+1) % n): 1 for i in range(n)})
sage: c = matrix(ZZ, n, {((i + 1) % n, i): 1 for i in range(n)})
sage: a = b - c
sage: a
[ 0  1  0  0 -1]
[-1  0  1  0  0]
[ 0 -1  0  1  0]
[ 0  0 -1  0  1]
[ 1  0  0 -1  0]
sage: a.parent()
Full MatrixSpace of 10 by 10 sparse matrices over Integer Ring