library('mgcv')
set.seed(1)
dat <- gamSim(1, n = 400, scale = 2, verbose = FALSE) ## 4 term additive truth

## add a factor for the random effect and include it's effect in the simulated data
fac <- as.factor(sample(1:10, 400, replace = TRUE))
dat$X <- model.matrix(~ fac - 1)
b <- rnorm(10) * 0.5
dat <- transform(dat, y = y + X %*% b)

## fit the model
m <- gam(y ~ s(fac, bs = "re") + s(x0), data = dat, method = "ML")

> coef(m)
(Intercept)    s(fac).1    s(fac).2    s(fac).3    s(fac).4    s(fac).5
 7.71917475  0.24762873  0.48122511 -0.17534482  0.03670067 -0.03800310
   s(fac).6    s(fac).7    s(fac).8    s(fac).9   s(fac).10     s(x0).1
-0.52197990  0.53661911 -0.23833765 -0.26088376 -0.06762439  0.02740419
    s(x0).2     s(x0).3     s(x0).4     s(x0).5     s(x0).6     s(x0).7
 0.15183166 -0.06347520 -0.15041864 -0.04918018  0.15609539 -0.03564953
    s(x0).8     s(x0).9
 0.78256472 -0.26510925

> head(model.matrix(m)[, 2:11])
  s(fac).1 s(fac).2 s(fac).3 s(fac).4 s(fac).5 s(fac).6 s(fac).7 s(fac).8
1        1        0        0        0        0        0        0        0
2        1        0        0        0        0        0        0        0
3        0        0        0        0        0        0        0        0
4        0        0        0        0        0        0        0        1
5        1        0        0        0        0        0        0        0
6        0        0        0        0        0        1        0        0
  s(fac).9 s(fac).10
1        0         0
2        0         0
3        0         1
4        0         0
5        0         0
6        0         0

> m[['smooth']][[1L]][['S']]
[[1]]
      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
 [1,]    1    0    0    0    0    0    0    0    0     0
 [2,]    0    1    0    0    0    0    0    0    0     0
 [3,]    0    0    1    0    0    0    0    0    0     0
 [4,]    0    0    0    1    0    0    0    0    0     0
 [5,]    0    0    0    0    1    0    0    0    0     0
 [6,]    0    0    0    0    0    1    0    0    0     0
 [7,]    0    0    0    0    0    0    1    0    0     0
 [8,]    0    0    0    0    0    0    0    1    0     0
 [9,]    0    0    0    0    0    0    0    0    1     0
[10,]    0    0    0    0    0    0    0    0    0     1